Introduction
Solving the Schrödinger equation in quantum mechanics is challenging, with exact solutions available only for a few simple models Consequently, researchers have invested significant effort into developing effective approximate methods, among which perturbation theory stands out This approach has been beneficial since the early days of quantum mechanics, offering analytical approximate solutions for various complex problems Perturbation theory not only aids in understanding physical phenomena but also resonates intuitively with many physicists, making it a popular choice for analysis.
The traditional textbook formulas for perturbation corrections can be complex and cumbersome for high-order calculations This book presents alternative strategies that facilitate both numerical and algebraic computations Our focus is on deriving exact perturbation corrections through straightforward, nontrivial models with practical applications Additionally, some of the algorithms explored here are applicable for numerical calculations, enhancing their versatility.
This book primarily focuses on methods that generate recurrence relations and mathematical algorithms, which are easy to calculate by hand and ideal for computer algebra systems For those exploring higher perturbation orders, utilizing computer algebra is essential Among various software options, we recommend Maple for its user-friendly interface, robust capabilities, reliability, and features that facilitate report writing and output conversion for word processing.
This chapter provides a concise overview of essential perturbation theory formulas in quantum mechanics, focusing on bound stationary states while also introducing time-dependent perturbation theory It is intended for readers already familiar with standard quantum mechanics concepts and notation Additionally, we will demonstrate practical applications of perturbation theory to stationary states in the continuum spectrum in Chapter 8.
Bound States
The 2s + 1 Rule
To accurately calculate energy corrections, it is essential to consider the eigenfunction corrections; however, we can derive all energy coefficients up to order 2s+1 from the eigenfunction corrections across all orders This approach utilizes more symmetric formulas, which we will outline briefly For instance, the matrix element < n,s |[E n, 1− ˆH ]| n,t > is relevant when s is less than t.
Using the general equation (1.6) we rewrite it as n,s
This process results in a reduction of the larger subscript and an increase in the smaller one, enhancing the symmetry of the matrix element We repeatedly apply this to equation (1.10) to achieve the most symmetric expression for the energy, which incorporates perturbation corrections of the lowest order to the eigenfunction For instance, the initial energy coefficients are derived from this method.
The symmetrized energy formulas, along with their generalizations, are well-established concepts that have been explored in greater detail by various authors.
Degenerate States
When unperturbed states are degenerate, the standard perturbation equations cannot be applied directly In cases where there are g linearly independent solutions to the unperturbed equation sharing the same eigenvalue, a different approach is required to analyze the system.
Hˆ0 n,a =E n, 0 n,a , a=1,2, , g n (1.22) we say that those states areg n -fold degenerate Any linear combination n, 0 g n a= 1
C a,n n,a (1.23) is an eigenfunction ofHˆ0 with eigenvalueE n, 0 Applying the bra< n,a | from the left to the equation of first order (1.7), we obtain an homogeneous system ofg n equations withg n unknowns: g n b= 1
As before we assume that< n,a | n,b >=δ ab and writeH a,b =< n,a | ˆH | n,b > Nontrivial solutions exist only if the secular determinant vanishes:
Theg n real rootsE n, 1 ,b , b=1,2, , g n are the corrections of first order for those states.
As we explore higher orders of perturbation theory, the notation becomes increasingly complex, making it less practical to continue along this path Therefore, we will defer a more systematic discussion of perturbation theory for degenerate states until Chapter 3.
InChapters 5and7we will show that it is sometimes convenient to choose a nonlinear perturbation parameterλin the Hamiltonian operator and expandH (λ)ˆ in a Taylor series aboutλ=0 as follows:
To effectively utilize perturbation theory, it is essential to solve the eigenvalue equation for the Hamiltonian operator \( \hat{H}_0 = \hat{H}(0) \) Once this equation is solved, the perturbation equations can be readily established and applied.
E n,j − ˆH j n,s−j , (1.27) and that the systematic calculation of the corrections is similar to that in preceding subsections.
Equations of Motion
Time-Dependent Perturbation Theory
The Schrödinger equation cannot be solved exactly, except for a few simple models, necessitating the use of approximate methods To apply perturbation theory, we express the Hamiltonian operator as Hˆ = Hˆ0 + λHˆ, where Hˆ0 is typically time-independent and Hˆ may vary with time Additionally, we factor the time-evolution operator to facilitate analysis.
U(t)ˆ = ˆU 0 (t)Uˆ I (t) (1.37) giving rise to the so-called interaction or intermediate picture [5] The time-evolution operator in the interaction pictureUˆ I is unitary and satisfies the differential equation ih¯ d dtUˆ I =λHˆ I Uˆ I , Hˆ I = ˆU 0 † Hˆ Uˆ0 (1.38)
The usual initial conditions areUˆ0 (0)= ˆ1 andUˆ I (0)= ˆ1.
ExpandingUˆ I in a Taylor series aboutλ=0
Uˆ I,j λ j (1.39) we obtain a recurrence relation for the coefficients [6]
Notice that any partial sum of the series (1.39) satisfies the initial conditionUˆ I (0)= ˆ1, but it is not unitary.
In certain instances, it is possible to select Uˆ0 to ensure that equations (1.39) and (1.40) yield an approximate expression for Uˆ I, which can be effectively utilized for calculating matrix elements and transition probabilities.
In order to illustrate the application of perturbation theory in the interaction picture we concentrate on the approximate calculation of operators in the Heisenberg picture when the Hamiltonian operator
If we expand a given Heisenberg operatorAˆ H in a Taylor series aboutλ=0
Aˆ H,j λ j , (1.41) then equation (1.35) withHˆ H = ˆHgives us ih¯ d dtAˆ H,j Aˆ H,j ,Hˆ0
We propose a solution to this operator differential equation of the formAˆ H,j = ˆU 0 † Bˆ j Uˆ0and derive a differential equation for the time-dependent operatorBˆ j ih¯dBˆ j dt = ˆU 0
If we define a dimensionless time variables=ωtin terms of a frequencyω, and a dimensionless
Hˆ = Hˆ hω¯ , (1.46) then we obtain a dimensionless Schrửdinger equation idUˆ ds = ˆHU ˆ (1.47)
Notice that we can derive equation (1.47) formally by settingh¯ =1 in equation (1.29).
One-Particle Systems
Most of this book is devoted to one-particle models because they are convenient illustrative examples More precisely, we consider a particle of massmunder the effect of a conservative force
F ( r )= −∇V ( r ), whereV ( r )is a potential-energy function andrdenotes the particle position The Hamiltonian operator for this simple model reads
2m +V ( r ) , (1.48) wherepˆ andrˆare vector operators with components(pˆ x ,pˆ y ,pˆ z )and(x,ˆ y,ˆ z), respectively Theyˆ satisfy the well-known commutation relations for coordinates and conjugate momenta u,ˆ pˆ v
The Hamiltonian operator is applicable to the relative motion of two particles with masses m1 and m2 In this context, the reduced mass is defined as m = m1 m2 / (m1 + m2), while the relative position is represented as rˆ = rˆ2 − rˆ1, and the relative momentum is expressed as pˆ = pˆ2 − pˆ1.
Mathematical equations are inherently dimensionless, leading us to consider the removal of dimensions from physical equations for simplification This process results in equations that are less cluttered by physical constants and parameters, allowing for a clearer understanding of the model's relevant parameters To achieve this, we introduce dimensionless variables: the coordinate \( q = \frac{r}{\gamma} \) and the momentum \( p = \frac{\gamma p}{h} \), where \( \gamma \) is an undefined unit of length Consequently, the Hamiltonian operator can be expressed in a more manageable form.
, v( q )= mγ 2 h¯ 2 V (γ q ) , (1.50) and we chooseγ in such a way that the form of the dimensionless Hamiltonian operatorHˆ mγ 2 H /ˆ h¯ 2 is as simple as possible.
In the case of the time-dependent Schrửdinger equation one also defines a dimensionless time s=ωt, as discussed earlier, and obtains i d dsUˆ = ˆHU,ˆ Hˆ |ˆp | 2
We obtain the dimensionless equation by formally settingh¯ =m=1 For brevity we writepinstead ofp when there is no room for confusion.
Examples
Stationary States of the Anharmonic Oscillator
As a first illustrative example we consider the anharmonic oscillator
2 +k xˆ M , M=4,6, (1.53) which in dimensionless form reads
In particular we chooseM=4 and apply perturbation theory withHˆ0=(pˆ 2 + ˆq 2 )/2, andHˆ = ˆq 4 The unperturbed problemHˆ0|n >= (n+1/2)|n >is nondegenerate and we easily calculate the matrix elementsH mn by means of the recurrence relation [7] m| ˆq j |n n
Notice that in this case equations (1.10) and (1.12) yield exact analytical results because< m| ˆH |n >
Using the Maple procedures outlined in the program section, we obtained the results presented in Table 1.1 It is important to note that the matrix elements and equal zero when λ=0, as they are precisely zero for the harmonic oscillator and emerge solely from the perturbation.
Harmonic Oscillator with a Time-Dependent Perturbation
This article demonstrates the application of time-dependent perturbation theory to a one-dimensional harmonic oscillator influenced by a simple time-dependent perturbation For perturbed harmonic oscillators, it is often useful to express dynamical variables using creation and annihilation operators, which adhere to the commutation relation [a, a†] = 1 The model Hamiltonian operator is represented as H = H₀ + λH, where H₀ is the unperturbed Hamiltonian and λH denotes the perturbation.
, H =f (t)aˆ+f (t) ∗ aˆ † , (1.56) f (t) is a complex-valued function of time, and f (t) ∗ its complex conjugate [8] The dummy perturbation parameterλis set equal to unity at the end of the calculation.
The dimensionless Schrửdinger equation idUˆ ds ˆ a † aˆ+1
U ,ˆ (1.57) wheres=ω 0 t, clearly reveals that the result will depend on the dimensionless functionf (t)/(hω¯ 0 ).
In order to facilitate comparison with earlier results, in this case we prefer to work with the original Schrửdinger equation.
It is our purpose to write the perturbation correctionsHˆ I,j in normal order (powers ofaˆ † to the left of powers ofaˆ) because it facilitates the calculation of matrix elements.
According to equation (1.40) the perturbation correction of first order is
Table 1.1 Perturbation Corrections for the Dimensionless Anharmonic
2 +λqˆ 4 Perturbation corrections to the energy of the nth excited state
First terms of the perturbation series for the ground state
A straightforward calculation shows that the correction of second order is
In order to obtain equation (1.62) notice that(β 3 )= −|β 1| 2 /2, where(z)stands for the real part of the complex numberz.
It is not difficult to verify that the time-evolution operator for this simple model is exactly given by [8]
Expanding the exponentials and keeping terms through second order inHˆ we obtain the results given above by perturbation theory.
On calculating the transition probabilities [8]
In the context of quantum mechanics, the transition probabilities \( P_{mn} \) can be expressed using the perturbation expression for \( U_I \), revealing that at first order, \( P_{mn} = 0 \) for \( |m - n| > 1 \), and at second order, \( P_{mn} = 0 \) for \( |m - n| > 2 \) By focusing on first-order perturbation corrections, we derive the standard selection rule for harmonic oscillators: \( |m - n| = \pm 1 \) However, employing the exact expression for \( U_I \) indicates that all transition probabilities are nonzero Notably, at short times, perturbation theory provides a reliable approximation, as the first-order correction \( P \) is proportional to \( |β_1| \), which approaches zero as time \( t \) approaches zero To further illustrate this concept, we will discuss a specific example.
Consider the periodic interaction given by f (t)=f 0cos(ωt) , (1.66) where|f 0| ¯hω 0for a weak interaction In the case of resonanceω=ω 0we have β 1 (t)= f 0
The outcome of this analysis is influenced solely by the dimensionless time variables ω₀t and the ratio f₀/(ℏω₀) Notably, the absolute values of the first and third terms on the right side of the equation remain small for all time values, whereas the absolute value of the second term exhibits a linear increase over time.
Perturbation theory yields reliable results when the parameter |β 1| remains small, specifically when |t| is much smaller than the time scale defined by ¯h/f 0 1/ω 0 This indicates that perturbation theory is applicable within a time interval significantly shorter than the harmonic oscillator's period, which is 2π/ω 0.
P 02 = |β 1| 4 /2P 01 = |β 1| 2 and the harmonic-oscillator selection rule is approximately valid.
Heisenberg Operators for Anharmonic Oscillators
In this section, we develop approximate expressions for Heisenberg operators specifically for anharmonic oscillators We confirm that the dimensionless time-evolution equation can be expressed as \( i\hat{U} ds = [\hat{H}, \hat{U}] \), with \(\hat{H}\) defined in equation (1.54).
In this article, we will use the notation pˆ for dimensionless momentum and t for dimensionless time To revert to the original units, it is essential to substitute x/γˆ for q, γˆ p/ˆ h¯ for ˆ p, and ωt for t throughout the text.
In the context of perturbation equations discussed earlier, the expressions Hˆ0 = (pˆ² + ˆq²)/2 and Hˆ = ˆq M serve as the foundational elements for analyzing cubic (M = 3) and quartic (M = 4) oscillators To effectively apply equation (1.45) recursively, it is essential to consider the established canonical transformations.
Table 1.2 presents results for second-order calculations with M = 3 and first-order calculations with M = 4 While the computation process is straightforward, it can be tedious, involving manual calculations of commutators followed by the use of Maple for integral calculations It is crucial to maintain the correct order of coordinate and momentum operators, as they do not commute For consistency, we have chosen to position powers of \( \hat{q} \) to the left of powers of \( \hat{p} \), adhering to the relation \( \hat{p} \hat{q} = -i + \hat{q} \hat{p} \).
Table 1.2 Perturbation Corrections to the Heisenberg Operatorqˆ H for
Perturbation Theory in the Coordinate
Introduction
In Chapter 1, we outlined a systematic approach to solving perturbation equations using a number representation that facilitates the calculation of essential matrix elements for equations (1.10) and (1.12) Alternatively, by expressing the Hamiltonian operator in the coordinate representation, the perturbation equations (1.6) transform into differential equations The unperturbed equation presents a solvable eigenvalue problem, while the perturbation corrections correspond to solutions of inhomogeneous differential equations This chapter explores several commonly employed strategies for addressing these equations.
The Method of Dalgarno and Stewart
The One-Dimensional Anharmonic Oscillator
As a first example we choose the widely discussed one-dimensional anharmonic oscillator
2 +λx 4 (2.6) that we split into a dimensionless harmonic oscillator and a quartic perturbationλx 4
Upon substituting the unperturbed ground state normalized to unity
Straightforward inspection reveals that the solutions are polynomial functions of the form
Substitution of equation (2.9) into equation (2.8) forj =1 leads to the polynomial equation
The eigenvalue equation does not determine the coefficientc 10that we derive from the normalization condition (cf equation (1.11))
We have derived the energy coefficient E1 without using equation (1.14) by ensuring that the solution meets the boundary condition, specifically that F1(x) and 0(x) are square integrable, which fully determines the energy.
By means of equations (1.16) and (1.19) we obtain the perturbation corrections of second and third order, respectively:
16 (2.15) that agree with the results ofTable 1.1.
Following this approach allows for the derivation of higher-order perturbation corrections To better illustrate this systematic calculation, we will utilize more intriguing and slightly more complex quantum-mechanical models The previously discussed simple anharmonic oscillator serves merely as an introductory example.
The Zeeman Effect in Hydrogen
The spinless hydrogen atom in a uniform magnetic field serves as a compelling example in both physics and mathematics This model is particularly engaging compared to the anharmonic oscillator and has been extensively analyzed using perturbation theory However, it presents greater mathematical challenges due to its non-separable nature, resulting in perturbation equations that involve two variables By aligning the z-axis with the magnetic field, the Hamiltonian operator can be expressed accordingly.
The equation 8mc²(x² + y²) (2.16) involves key parameters such as the atomic reduced mass (m), the charge of the electron (e), the speed of light (c), the z-component of the angular-momentum operator (Lˆ z), and the magnitude of the magnetic induction (B) [13].
If we define units of lengthγ = ¯h 2 /(me 2 )and energyh¯ 2 /(mγ 2 )=e 2 /γwe obtain a dimension- less Hamiltonian operator
In the equation (2.17), the dimensionless operator Lˆ z is expressed in units of h¯, while the perturbation parameter λ is defined as B 2 h¯ 6 /(8m 4 c 2 e 6 ) To avoid confusion, we use the same symbols for both original and dimensionless quantities To revert to the original units after calculations, lengths, energy, linear momenta, angular momenta, and wavefunctions must be multiplied by γ, e 2 /γ, h/γ, h¯, and γ − 3 / 2, respectively Given that the commutation relation [ ˆH,Lˆ z ] = 0 holds, we exclude the constant of motion Lˆ z from the perturbation calculation and subsequently incorporate the eigenvalue of √.
2λLˆ z to the resulting energy In other words, from now on, we considerHˆ −√
We can solve the perturbation equations in several different coordinate systems Here we choose a kind of modified spherical coordinates in whichu=cos(θ)takes the place ofθ: x =r
By straightforward application of the general method outlined inAppendix A, we obtain the form of the laplacian∇ 2 in terms of such coordinates, and the perturbation equations for the factor functions
Notice that∂F j /∂φ =0 because the perturbation is independent ofφ, and the state depends onφ only through 0which is an eigenfunction ofLˆ z
In what follows we call ground state the one that correlates with the state 1sof the hydrogen atom as
Despite the established understanding that states with negative magnetic quantum numbers can attain lower energy levels at sufficiently high magnetic field strengths, the dimensionless ground state energy and eigenfunction remain critical for analysis.
Because 0is independent of u andφ, the perturbation equations (2.20) take a simpler form:
Proceeding exactly as in the preceding example, at each stepj =1,2, we substitute a poly- nomial solution of the form
To calculate the coefficients \( c_{jik} \), substitute \( 3j \) into equations (2.22) and (2.23) The initial perturbation corrections can be computed manually with ease, but as the perturbation order increases, the complexity rises significantly, making computer algebra a preferable choice In the program section, we present a series of straightforward Maple procedures designed for the systematic calculation of perturbation corrections to the ground state Results are summarized in Table 2.1.
Table 2.1 Perturbation Corrections to the Ground State of a Hydrogen Atom in a Magnetic Field by Means of the Method of Dalgarno and Stewart
Logarithmic Perturbation Theory
The One-Dimensional Anharmonic Oscillator
In examining the anharmonic oscillator, we find that its logarithmic derivative represents a scalar in this one-dimensional scenario Analyzing the perturbation equations reveals that the relationship between the functions can be expressed as f_j - j_i = 0 f_i f_j - i + x^2 δ_j 0 + 2x^4 δ_j 1 - 2E_j = 0, leading to the conclusion that f_j can be defined as f_j = c_ji x^2 i + 1 This insight into the behavior of the anharmonic oscillator is crucial for understanding its dynamics.
For example, an appropriate solution to the unperturbed equation f 0 −f 0 2 +x 2 −2E 0=0 (2.35) isf 0=x,E 0=1/2 At first order f 1 −2f 0 f 1+2x 4 −2E 1=0 (2.36) we try f 1=c 10 x+c 11 x 3 (2.37) and obtain f 1= 3x
4x 4 (2.39) agrees with the functionF 1 (x), equation (2.13), given by the method of Dalgarno and Stewart if c 1=9/16.
We do not proceed with the discussion of higher perturbation corrections for this simple model and turn our attention to the Zeeman effect in hydrogen.
The Zeeman Effect in Hydrogen
In this section, we analyze the dimensionless Hamiltonian operator for a hydrogen atom subjected to a magnetic field By employing the coordinates previously established in the Dalgarno and Stewart method, we derive the corresponding perturbation equations.
By inspection we conclude that
2 j+ 1 k= 0 c jki r k u 2 i (2.41) for the ground state For simplicity we arbitrarily choose the normalization constantsc j 00equal to zero.
The algorithm for calculating perturbation corrections closely resembles that of the Dalgarno and Stewart method, making it unnecessary to present the Maple program here The logarithmic perturbation theory program is more efficient, running faster and using less memory, allowing for the computation of additional perturbation corrections The first eight perturbation corrections to the energy, which are not included in Table 2.2, align with those found in Table 2.1 Additionally, Table 2.2 provides the normalization constants c1 and c2, which are essential for deriving F1 and F2 from G1 and G2, as outlined in equations (2.32).
The Method of Fernández and Castro
The One-Dimensional Anharmonic Oscillator
The time-independent Schrửdinger equation for a dimensionless one-dimensional quantum-me- chanical model
& (q)+2[E−V (q)]&(q)=0 (2.53) is a particular case of (2.42) withP =1,Q=0, andR =2(E−V ) Given a closely related exactly solvable problem of the form
As a simple nontrivial example we choose the anharmonic oscillator
Both \( V_0(q) \) and \( V(q) \) exhibit definite parity, which means they are either even or odd Consequently, \( A(q) \) must be even while \( B(q) \) is odd to ensure that \( \phi_0 \) and \( \phi \) maintain the same parity A simple analysis of equations (2.57) confirms this conclusion.
A k (q) α k i= 0 a ki q 2 i , B k (q) β k i= 0 b ki q 2 i + 1 , (2.58) whereα 1=β 1=1, α k =β k− 1+3, andβ k =α k− 1+1 for allk >1.
The perturbation equations do not specify the coefficients \( a_k^0 \), which we determine through normalization, opting for an intermediate normalization where \( a_k^0 = 0 \) If needed, we can normalize the eigenfunction later Table 2.3 presents results based on \( E_0 = n + 1/2 \), with \( n = 0, 1, \ldots \) representing the harmonic oscillator quantum number Additionally, we have derived numerous perturbation corrections using the straightforward program detailed in the program section.
Table 2.3 Method of Fernández and Castro for the Anharmonic Oscillator
The method developed by Fernández and Castro provides efficient general expressions for eigenfunctions and eigenvalues based on the quantum numbers of the unperturbed problem, outperforming the Dalgarno and Stewart method and logarithmic perturbation theory in speed and memory usage This efficiency stems from the simplicity of the polynomial functions A_k(x) and B_k(x), which handle most details of the perturbed eigenfunction, including oscillations in the classical region However, a notable limitation of the Fernández and Castro method is its less straightforward application to nonseparable models.
The method of Fernández and Castro has already been applied to several separable quantum- mechanical models [21]–[23] We will mention some of them later in this book.
Introduction
In the previous chapters, we approximated the time-independent Schrödinger equation by calculating perturbation corrections to both eigenvalues and eigenfunctions The process of determining perturbation corrections for eigenfunctions is often more complex and time-consuming Therefore, if the primary focus is on energy, it may be beneficial to avoid detailed analysis of the eigenfunction This article explores several established strategies for this purpose, including two methods that utilize expectation values and moments in place of the eigenfunction, alongside a third approach that employs true perturbation theory, relying solely on the Hamiltonian operator.
Hypervirial and Hellmann–Feynman Theorems
(3.1) for any pair , of vectors of the state space In particular, the Hamiltonian operator satisfies equation (3.1), and in the case that is an eigenfunction ofHˆ with eigenvalueE,Hˆ =E, we have
IfWˆ is an arbitrary linear operator such that = ˆWthen(Hˆ−E)= ˆW(H−E)+[ ˆH ,Wˆ] , and equation (3.2) leads to the hypervirial theorem [25]
If is normalized to unity< | >=1 we simply write
=0 (3.4) where< >denotes the quantum-mechanical expectation value.
When the Hamiltonian operator is influenced by a parameter λ—such as particle charge, mass, field strength, or an arbitrary variable for perturbation theory—the eigenvalues and eigenfunctions of the Hamiltonian will also vary with that parameter Differentiating with respect to λ reveals the relationship between these changes and the underlying physical properties.
∂λ =0 (3.5) and take into account equation (3.2), then we obtain the Hellmann–Feynman theorem [25]
(3.7) if is normalized to unity.
In what follows we show that the general results just derived facilitate the application of pertur- bation theory.
The Method of Swenson and Danforth
One-Dimensional Models
We illustrate the main ideas behind the method of Swenson and Danforth by means of a one- dimensional model
First, rewrite the commutator[ ˆH,Wˆ]for the linear operator
= ˆfpˆ+ ˆg , (3.9) wherefˆ=f (x), andˆ gˆ=g(x), in the following way:ˆ
Second, choose the arbitrary functionsf (x)ˆ andg(x)ˆ so that the coefficient ofpˆin equation (3.10) vanishes: ˆ g=1 2 f ,ˆ pˆ
, (3.11) and writepˆ 2 in terms ofHˆ andVˆ Equation (3.10) becomes
Finally, the hypervirial theorem (3.4) gives us an expression for expectation values of operators that commute with the coordinate operator:
Notice, for example, that[ ˆp,fˆ]is a function only of the coordinate operator as it follows from the Jacobi identity p,ˆ ˆ x,fˆ + f ,ˆ ˆ p,xˆ + ˆ x, f ,ˆ pˆ ˆ x, f ,ˆ pˆ
Equation (3.13) reduces to the well-known virial theorem for the particular casefˆ= ˆx:
In the coordinate representationpˆ= −ihd/dx, and equation (3.13) reduces to¯
To derive equation (3.16), we denote the derivative with respect to x using a prime notation For simplicity, we omit the caret on the functions of the coordinate operator, indicating that x(x)ˆ is equivalent to x(x) While it may be easier to derive equation (3.16) directly in the coordinate representation, we have chosen a more complex approach to ensure that the main equation (3.13) is independent of any specific representation.
Equation (3.16) enables one to obtain expectation values in terms of the energy of a stationary state of a given exactly solvable model As an example consider the dimensionless harmonic oscillator
Choosingf (x) = x 2 j+ 1 , j = 0,1, , equation (3.16) withh¯ = m = 1 becomes a three-term recurrence relation that we conveniently rewrite as
In our analysis, we exclude the expectation values that are zero due to the properties of the eigenfunctions being either even or odd To derive all the expectation values X_j for j = 1, 2, , we utilize equation (3.18) and recognize that X_0 equals 1.
X j is a polynomial function ofEof degreej, whereE=n+1/2, andn=0,1, is the quantum number.
In cases where the quantum-mechanical model cannot be solved exactly, perturbation theory is utilized to analyze the recurrence relation for expectation values This method is exemplified through the study of a simple anharmonic oscillator.
+λx 2 K , (3.20) whereK =2,3, andλis a perturbation parameter The recurrence relation for the expectation values becomes
If we expand the expectation values and the energy in a Taylor series aroundλ=0
X j,i λ i , (3.22) then equation (3.21) gives us a recurrence relation for the coefficientsE i andX j,i Substitutingj−1 forj it reads
X 0 ,i =δ 0 i (3.24) as follows from the normalization conditionX 0=1.
The recurrence relation (3.23) gives us the perturbation correctionsX j,i in terms of the yet unknown energy coefficientsE i The Hellmann–Feynman theorem
∂λ =X K (3.25) provides an additional equation that enables us to solve the problem completely Expanding equa- tion (3.25) in a Taylor series aboutλ=0 we have
E i = 1 iX K,i− 1 (3.26) for alli >0 By straightforward inspection of the recurrence relation (3.23) one easily convinces oneself that it is sufficient to calculate the coefficientsX j,i , for alli =1,2, , p−1 andj 1,2, , (p−i)(K−1)+1 in order to obtainE p
Table 3.1 Method of Swenson and Danforth for the Anharmonic Oscillator
Note:The energy coefficients are identical to those inTable 2.3.
This section presents a series of straightforward Maple procedures designed to streamline the calculation of perturbation corrections for the anharmonic oscillator Table 3.1 displays the perturbation corrections \(E_i\) and \(X_{1,i}\) for \(K=4\), which align with our earlier findings in Chapter 2 By utilizing the perturbation corrections to the energy and equation (3.26), one can effortlessly derive the coefficients \(X_{2,i}\).
The method of Swenson and Danforth also applies to potentials that are not parity invariant. Consider, for example, the cubic-quartic anharmonic oscillator
In this case we choosef (x)=x j ,j =0,1, , and define
Repeating the algebraic steps in the discussion of the preceding example we obtain the recurrence relation
(3.29) from the hypervirial theorem, and
(3.30) from the Hellmann–Feynman theorem The initial conditions are also given by equation (3.24).
In order to obtainE p we need the perturbation coefficientsX j,i for alli =0,1, , p−1, and j =1,2, ,2(p−i+1).
Table 3.2 presents the perturbation corrections to energy and specific moments It is important to note that when α equals 0, the energy coefficients align with those found in Table 2.3, leading to the disappearance of all X1,i due to the parity-invariance of the resulting potential-energy function.
Central-Field Models
InChapters 1and2we briefly outlined the Hamiltonian operator for one- and two-particle systems.
This chapter focuses on central conservative forces defined by the gradient of a potential-energy function V(r), where r represents the distance between particles The Schrödinger equation can be separated in spherical coordinates (r, θ, φ), allowing us to express the eigenfunctions of the Hamiltonian operator as nlm(r, θ, φ) = Rnl(r)Ylm(θ, φ) Here, n, l, and m are quantum numbers that uniquely determine the state, with Ylm representing the spherical harmonics We will now utilize the dimensionless form of the Schrödinger equation from Chapter 1, concentrating on deriving the radial factor R(r) of the eigenfunction, defined as (r) = rR(r), which acts as an eigenfunction of the one-dimensional-like Hamiltonian operator.
The bound states satisfy the boundary conditions
If we naively follow the steps of the discussion above regarding the hypervirial theorem for actual one-dimensional models, and choose the functionf (r)=r j , we obtain
The hypervirial theorem's validity is limited by boundary conditions, particularly at r = 0 While the theorem's formulation under various boundary conditions is established, we will briefly discuss this aspect for the sake of thoroughness.
(r) ∗ χ(r)dr , (3.34) where(0)=0 The radial Hamiltonian (3.31) satisfies
Table 3.2 Method of Swenson and Danforth for the Anharmonic Oscillator
− 50820705 512 E 0− 20771985 64 E 0 3− 4402125 32 E 0 5 α 2 − 10195983 1024 as follows from integration by parts For the particular caseχ= ˆD,Dˆ =d/dr, we have
As r approaches zero, if the limit of r squared times V(r) equals zero, the eigenfunctions of the Hamiltonian operator exhibit a behavior characterized by Cr^(l+1) near r = 0, where C is a nonzero constant It is important to note that the master equation does not apply for states with l = 0 when j = 0, as the function evaluates to zero in this case However, this issue is resolved for states where l is greater than zero.
Another important point to notice is that the integrand in the expectation value r j
The expression (3.38) approaches the form |C| 2 r 2 l+j+ 2 near the origin, indicating that the expectation value is undefined for j < −2l−2 This consideration is crucial when dealing with perturbations that exhibit singular behavior at the origin.
In what follows we consider two exactly solvable models as illustrative examples The master equation (3.33) for the harmonic oscillator in three dimensions
According to equation (3.36), forj =0 we have l(l+1)R − 3−R 1= −1
2(0) 2 , (3.42) and forj =2 equation (3.40) reduces to
To accurately determine the expectation values of odd powers of the radial coordinate in the context of the Schrödinger equation, additional information about the solutions is necessary However, equation (3.40) allows for the calculation of \( R^2_j \) for all \( j > 0 \) based solely on energy considerations.
2, ν=0,1, (3.45) into the Hellmann–Feynman theorem
Taking into account this expression and the master equation (3.40), we easily calculate the expectation valuesR 2 j for all−(l+1)≤j 0 for alljwe clearly see that this expression is not valid whenl=0 in agreement with the discussion above Analogously, it follows from equation (3.40) withj = −3 that
We now consider anharmonic oscillators with potential-energy functions
2 +λr 2 K , K =2,3, (3.50) as illustrative examples In order to apply the method of Swenson and Danforth, we take into account the perturbation series for the expectation values
R j,i λ i (3.51) and proceed as in the one-dimensional case obtaining the recurrence relation
The normalization condition and the Hellmann–Feynman theorem give usR 0 ,j =δ 0 j and
E i = 1 iR K,i− 1 , (3.53) respectively In order to obtain the perturbation correction to the energyE p , we need all the coeffi- cientsR j,i withi=0,1, , p−1 andj =1,2, , (p−i)(K−1)+1.
Table 3.3 Method of Swenson and Danforth for the Anharmonic Oscillator
Table 3.3 presents the perturbation corrections to energy and R3, but we refrain from providing additional results as the length of these corrections increases significantly with higher perturbation orders A straightforward Maple program can generate numerous additional perturbation corrections, though we do not include it here as it resembles the previously discussed program for the one-dimensional model.
Another interesting exactly-solvable model is the nonrelativistic hydrogen atom Upon solving the dimensionless Schrửdinger equation with the dimensionless Coulomb interaction between electron and nucleus
V (r)= −1 r (3.54) one obtains the dimensionless energy eigenvalues
2n 2 , n=ν+l+1, (3.55) whereν=0,1, , n−l−1 The master equation (3.33) reduces to
R j− 3+(2j−1)R j− 2=0 (3.56) Whenj =1 we obtain the well-known virial theorem
The Hellmann–Feynman theorem (3.46) gives us
From the master equation (3.56) withj =0 we obtain
This article examines simple perturbations represented by the form λr K, where K is greater than zero The methodology is also applicable to singular perturbations (K less than zero), with specific considerations regarding the expectation values of negative powers of the radial coordinate We will substitute the potential-energy function accordingly.
V (r)= −1 r +λr K (3.60) into the master equation (3.33) we obtain the recurrence relation
Expanding the energy and expectation values in Taylor series aboutλ=0 as in preceding examples, equation (3.61) gives us the recurrence relation
In this analysis, we utilize the Hellmann–Feynman theorem to demonstrate that equation (3.53) is applicable to our example However, the recurrence relation (3.62) proves inadequate for calculating perturbation corrections to R − 1 due to a vanishing denominator To address this issue, we apply the virial theorem derived from equation (3.61) with j = 1, and subsequently expand the resulting expression using a Taylor series.
R − 1 ,i = −2E i +(K+2)R K,i− 1 (3.63)Table 3.4shows sample results: one easily obtains more perturbation corrections by means of a simple Maple program written according to equations (3.53), (3.62), (3.63), and the normalization conditionR 0 ,j =δ 0 j
More General Polynomial Perturbations
The Swenson and Danforth method extends to a broader range of polynomial perturbations beyond those previously mentioned Readers can derive appropriate perturbation equations by utilizing the hypervirial and Hellmann–Feynman theorems for various scenarios.
Table 3.4 Method of Swenson and Danforth forHˆ = −∇ 2
Moment Method
Exactly Solvable Cases
WhenH i,j =0 for alli > jthe eigenvalue equation (3.69) becomes an exactly solvable triangular system of homogeneous linear equations
In the nondegenerate case where H i,i = H j,j for all i ≠ j, if E equals H j,j for all j, then all moments F j will vanish, resulting in the null vector due to the completeness of the set {f j} Therefore, for a nontrivial solution to exist, E must correspond to one of the diagonal coefficients.
All the nonzero moments are proportional toF n which we may arbitrarily choose equal to unity as an intermediate normalization condition.
As an illustrative example consider the dimensionless harmonic oscillator
(3.73) and the set of functions f j =x 2 j+s exp
In this case we have
2 f j (3.75) so that the only nonzero coefficientsH i,j are
The argument above leads to the well-known energy eigenvaluesE=(2j+s+1/2), wheres=0 ands=1 apply to even and odd states, respectively The recurrence relation for the moments is
The fully degenerate caseH i,i =H j,j for alli,j is also interesting ChoosingE =H j,j , the eigenvalue equation (3.71) becomes
If all the coefficientsH j− 1 ,j are nonzero, then all the momentsF j are zero, and is the null vector.
We must therefore assume thatH n− 1 ,n =0 The nonzero moments given by the recurrence relation
H i,j + 1 F i , j =n, n+1, (3.79) are proportional toF n− 1which we may arbitrarily choose equal to unity.
A simple example of full degeneration is given by the radial Hamiltonian operator for the Coulomb problem
Choosing the functions f j =r j+l+ 1 exp(−αr), j=0,1, (3.81) the only nonzero Hamiltonian coefficients are
Therefore, fromE=H n,n andH n− 1 ,n =0 we obtain the well-known results α= 1
2N 2 , (3.83) whereN =n+l+1 is the principal quantum number of the hydrogen atom [40].
Perturbation Theory by the Moment Method
If the moment equations (3.69) are not exactly solvable, then we apply perturbation theory We introduce a dummy perturbation parameterλas follows:
H m,j f m , (3.84) so that the recurrence relation for the moments becomes j m= 0
Notice that we have split the recurrence relation (3.69) into a solvable part and a perturbation that vanish whenλ=0 We can therefore try approximate solutions in the form of Taylor series about λ=0:
The normalization condition can be flexible, with F n often set to 1; however, in certain instances, a more complex normalization approach proves to be more effective This is illustrated by rewriting equation (3.85).
Second, look for a set of coefficientsC j ,j =0,1, , n, such that n j= 0
It is convenient to choose the coefficientsC j to be solutions to the homogeneous linear system of equations n j=m
In the case of a nondegenerate problem, there exists a unique linearly independent set of coefficients, denoted as C j Conversely, if the problem is degenerate, multiple linearly independent sets of coefficients will be present We will demonstrate both scenarios with relevant examples.
Once we have the coefficientsC j , we rewrite the equation for the energy as follows
C j F j =1 (3.93) leads to a useful expression for the energy in terms of the moments
Nondegenerate Case
Suppose thatH j,j = H n,n for all j = n Upon expanding the energy and moments in equa- tion (3.85) in Taylor series aboutλ=0 and solving forF j,i we obtain
Notice that this equation does not give us the perturbation corrections to the momentF n , which we obtain from the arbitrary intermediate normalizationF n =1 that leads to
Substitution of this normalization condition into equation (3.88) forj =nyields an expression for the energy
H m,n F m , (3.97) and, consequently, for its perturbation corrections
Remember thatF j, 0=0 for allj < nas shown above for the chosen exactly solvable models.
We easily obtain exact expressions for the perturbation corrections whenH i,j =0 for all|i−j| larger than some positive integer As an example consider the dimensionless anharmonic oscillator
+λx 4 (3.99) and the set of functions (3.74) The recurrence relation for the moments is
2 F j− 1+2(j−n)F j −-EF j +λF j+ 2=0 (3.100) and the intermediate normalization conditionF n =1 leads to
Expanding the energy and moments in Taylor series aboutλ=0, we obtain
The provided equations, along with the normalization condition, allow us to determine all perturbation corrections to energy and moments To compute the energy correction \(E_p\), it is essential to calculate the moment coefficients \(F_{j,i}\) within the specified ranges: for \(i=0,1,\ldots,p-1\), the coefficients should satisfy \(n-2i \leq j \leq n+2(p-i)\), and for \(i=p\), the range is \(n-2p \leq j \leq n-1\).
Table 3.5 presents the results for the first four states of the anharmonic oscillator, obtained through a basic Maple program Although the program is not displayed due to its slower performance and less comprehensive results compared to the previously outlined methods by Swenson and Danforth, Table 3.5 serves as a useful reference for readers to verify their own calculations.
Table 3.5 Moment-Method Perturbation Theory for the Anharmonic Oscillator
Another interesting example is the radial Hamiltonian operator for a perturbed Coulomb problem
2r 2 −1 r +λr K , (3.104) whereK=1,2, Choosing the set of functions (3.81) the recurrence relation for the moments is
2 F j− 2+ j −n n+l+1F j − 1−-EF j +λF j+k =0, (3.105) where we have substituted the value ofαgiven by equation (3.83) and
The choicen=0 selects the states with angular and principal quantum numbersl =0,1, , and
N =l+1, respectively, commonly denoted 1s, 2p, 3d, that are free from radial nodes When j =0 we obtain
-EF 0=λF K , (3.107) so that the intermediate normalization conditionF 0=1 leads to
We obtain an expression forF − 1from the general equation (3.105) withj =1:
The expansion of the energy and moments in Taylor series aboutλ=0 leads to
The last expression follows from substitutingj+1 forjin equation (3.105) The energy coefficients are given by
In order to obtainE p one has to calculate all the moment coefficientsF j,i withi=1,2, , p−1,
The choicen = 1 selects all the states withN = l+2: 2s, 3p, 4d, ., and their treatment illustrates the application of equations (3.92)–(3.94) The general equation (3.105) withn = 1 yields
In this section, we present the equation \((2l+3)F_0 + F_1 l + 2 - EF_2 + \lambda F_K + 2 = 0\) for \(j = 0, 1, 2\), highlighting the challenges in developing a comprehensive system of equations for general cases To illustrate the process, we provide a specific example and encourage readers to extrapolate the strategy for other scenarios Additionally, we will explore more illustrative examples throughout the book to offer further insights In the current problem, we multiply equation (3.114) by \((l+1)(l+2)\) and subtract it from equation (3.115), effectively eliminating \(F_{-1}\) and deriving a preliminary expression for the energy.
F 1−(l+1)(l+2)F 0=1 (3.118) leads to a simpler expression for the energy that is suitable for the application of perturbation theory:
From equations (3.116) and (3.118) we derive the following expression forF 0:
Expanding the energy and moments in Taylor series about λ = 0 we obtain all the necessary recurrence relations for the perturbation coefficients
The calculation is analogous to the one above except that in this case 2≤j ≤(p−i)(K+1)−1. Table 3.6shows perturbation coefficients for the energy and arbitrarily selected moments when
K = 1 The energy coefficients agree with those inTable 3.4 provided that we substitute the appropriate value ofE 0in each case: E 0= −1/[2(l+1) 2 ]andE 0= −1/[2(l+2) 2 ]for the states with zero and one radial node, respectively.
Degenerate Case
In order to illustrate the application of the moment method to a model with degenerate unperturbed states, we choose a simple nontrivial anharmonic oscillator in two dimensions with dimensionless Hamiltonian operator
, (3.125) whereλ,a,b, andcare real and positive.
The potential-energy functionV (x, y)is a single infinite well with a minimumV =0 at origin.
It is invariant under the transformation(x, y)←→(−x,−y), and in the particular case thata=b, it is also invariant under the exchange(x, y) ←→ (y, x) The latter higher symmetry makes the
Table 3.6 Moment-Method Perturbation Theory for the Perturbed Coulomb Model
States with No Radial Nodes
States with One Radial Node
The moment method simplifies the application of (2l+3) by treating degenerate states similarly to nondegenerate ones However, in this study, we do not limit ourselves to this simplification and aim to maintain the generality of our results.
The set of functions f ij =x i y j exp
, i, j=0,1, (3.126) enables us to construct the recurrence relation for the moments
To effectively apply perturbation theory, it is essential to clearly distinguish the functions and moments involved, as indicated by the use of two subscripts in equations (3.126) and (3.127) This approach enhances clarity compared to the previous general case discussions, where only a single subscript was utilized The moment recurrence relation is a key component in this context.
The unperturbed energy is defined as E₀ = N + 1 Due to the even-number displacement of subscripts in equation (3.128), there are four distinct sets of solutions identified as (e,e), (e,o), (o,e), and (o,o), where 'e' represents even parity and 'o' denotes odd parity These solutions align with the symmetry classes of the eigenfunctions.
The coefficients of the perturbation series
In this article, we demonstrate how to apply perturbation corrections to the lowest states of the anharmonic oscillator by utilizing an appropriate expression for energy.
For the ground state we chooseN =0 Settingi=j =0 into equation (3.128) we obtain
The intermediate normalization conditionF 0 , 0=1 determines the coefficients
F 0 , 0 ,m =δ 0 m (3.133) that we cannot obtain from the general recurrence relation (3.131), and also gives us a simple expression for the energy coefficients:
Notice that this state belongs to the class (e,e) mentioned above In order to obtainE p we need the moment coefficientsF i,j,m for allm=0,1, , p−1,i, j=0,2, ,4(p−m).
WhenN =1 we identify two unperturbed states with the same energy and different symmetry. Settingi=0, andj =1 we have
The intermediate normalization conditionF 0 , 1=1 leads to
In order to obtainE p we needF i,j,m for allm = 0,1, , p−1,i = 0,2, ,4(p−m), and j =1,3, ,4(p−m)+1 This state belongs to the class (e,o).
The remaining state forN =1 belongs to the class (o,e) Arguing as in the preceding case we obtain
In order to obtainE p we needF i,j,m for allm=0,1, , p−1,i=1,3, ,4(p−m)+1, and j =0,2, ,4(p−m).
For the case of N = 1, the unperturbed states are treated as nondegenerate due to the lack of coupling from perturbations, despite their inherent degeneracy The moment method application highlights the independence stemming from the symmetry of these states Specifically, the eigenfunctions 0,1(x,y) and 1,0(x,y) of the Hamiltonian Hˆ exhibit distinct symmetry properties: 0,1(−x,y) equals 0,1(x,y), while 0,1(x,−y) equals -0,1(x,y), and similarly, 1,0(−x,y) equals -1,0(x,y).
The moments with subscripts (o,e) vanish for N = 0 and 1, while those with subscripts (e,o) vanish for other states, highlighting the symmetry embedded in the chosen functions f i,j without requiring explicit consideration of eigenfunctions The states N = 0 and N = 1 do not introduce new insights compared to earlier one-dimensional problems, aside from an additional subscript in the moments In contrast, states with N = 2 present a more complex scenario, as the denominator in equation (3.131) approaches zero when i + j = N, allowing for N + 1 degenerate unperturbed states This results in N + 1 linearly independent solutions, with only one state for N = 0 and two states for N = 1, categorized into (e,o) and (o,e) classes Generally, for even N, degenerate states fall into (e,e) or (o,o) categories, while for odd N, they belong to (o,e) or (e,o) For N = 2, there are three states, beginning with the (o,o) case.
Settingi=j =1 in equation (3.128) we obtain
, (3.140) which suggests the intermediate normalization conditionF 1 , 1=1 that leads to
To obtain E_p, it is necessary to consider all F_{i,j,m} where m ranges from 0 to p−1 and i, j take values of 1, 3, , 4(p−m)+1 The remaining two states are classified as (e,e), leading to perturbation coupling that must be treated explicitly as degenerate Specifically, when (i, j) equals (0,0), (2,0), and (0,2), the results are as follows.
= 0, (3.145) respectively Subtracting twice equation (3.144) from equation (3.143) gives
=0, (3.146) and subtracting twice equation (3.145) from equation (3.143) yields
We arbitrarily choose the intermediate normalization condition
Substituting equation (3.149) into equation (3.146) and dividing by λ we derive another useful equation a
We calculate the perturbation corrections toF 0 , 0from equation (3.143):
(3.151) and the perturbation corrections toF 0 , 2from the intermediate normalization condition (3.148):
The general recurrence relation (3.131) withN =2 provides all the remaining moment coefficients
In this analysis, we consider the coefficients F i,j,m, excluding F 2,0,m, derived from equation (3.150) We expand all moments in a Taylor series around λ = 0 and gather the coefficients for each power of λ Utilizing equations (3.131), (3.151), and (3.152), we express each moment coefficient F i,j,m in relation to previously calculated coefficients and the unknown F 2,0,m, which we subsequently determine Notably, the zero-order equation is quadratic in nature.
4cF 2 2 , 0 , 0 +18(b−a)F 2 , 0 , 0−c=0 (3.153) and admits two real roots
The equation 81(b−a)² + 4c² (3.154) leads to two sets of moment coefficients F i,j,m, representing the two independent solutions previously discussed The existence of multiple solutions was expected, as highlighted in the general equation (3.91) For perturbation orders exceeding zero, the equations become linear concerning the unknown moment coefficient F 2,0,m; however, these equations are complex and not pertinent to the current discussion.
We finally obtain the energy coefficients from
Unlike the conventional approach to degenerate states, where the energy coefficient is derived from a secular determinant, this case involves a moment that emerges from what appears to be a secular equation.
This section presents straightforward Maple procedures for implementing the moment method based on the previously discussed equations Sample results for the states analyzed are displayed in Table 3.7.
The moment method has been effectively utilized to create renormalized perturbation series for the energies of two-dimensional anharmonic oscillators, providing highly accurate results that are easily comprehensible through the theoretical framework and discussions presented in Chapter 6 Additionally, this method has been applied to coupled Morse oscillators by expanding the potential-energy function in a Taylor series around its minimum, resulting in a perturbation series that converges for all states examined Chapter 7 will further explore the application of perturbation theory to nonpolynomial potential-energy functions using a straightforward polynomial approach.
Relation to Other Methods: Modified Moment Method
The moment method is a versatile strategy that can be simplified into other techniques under specific conditions Notably, when the complete set of vectors {f j} is orthonormal, the moment method aligns with the standard textbook approach previously mentioned.
The moment method effectively demonstrates the hypervirial theorem By selecting the vector \( \hat{W} \), which is a linear operator and an eigenfunction of \( \hat{H} \), we can derive the theorem using the same reasoning outlined in Section 3.2.
The moment method of perturbation theory does not provide energy corrections based on the quantum number of the unperturbed model, making it less attractive compared to the Swenson and Danforth method for addressing separable systems discussed in Chapter 1.
Table 3.7 Moment Method for the Two-Dimensional Anharmonic Oscillator
Table 3.7 (Cont.)Moment Method for the Two-Dimensional Anharmonic Oscillator
Table 3.7 (Cont.)Moment Method for the Two-Dimensional Anharmonic Oscillator
The combination of the moment method with Fernández and Castro's approach, as discussed in Chapter 2, effectively addresses the limitations of the 81(b−a)² + 4c² models This is demonstrated through a straightforward one-dimensional problem.
2 d 2 dx 2 +V 0 (x) (3.157) Choosing a function of the form
F (x)=A(x) 0 (x)+B(x) 0 (x) , (3.158) whereA(x)andB(x)are two differentiable functions, and 0 (x)is an eigenfunction ofHˆ0,
2 +(V 1−-E) B , (3.161) then the term containing 0 vanishes and
=0, (3.162) which is the master equation of the modified moment method.
As a particular example consider the anharmonic oscillator
SubstitutingN−1 forN and expanding in Taylor series aboutλ=0, we obtain
In order to derive an expression for the energy we chooseA=1 andB =0, which are consistent with equation (3.161) and lead to
Therefore, the intermediate normalization conditionF 0=1 leads to
In order to obtainE p we have to calculateF N,i fori=0,1, , p−1,N =1,2, , (p−i)K.
Calculating the first perturbation corrections by hand is straightforward, but as the perturbation order increases, the process becomes tedious, necessitating the use of computer algebra We have developed a simple Maple program, similar to the one for the Swenson and Danforth method, though it is not included here Table 3.8 presents results for the quartic anharmonic oscillator (K = 2), which may be useful for testing purposes Since the perturbation corrections to the energy are identical to those in Table 2.3, we focus on displaying some moment coefficients.
Table 3.8 Modified Moment Method for the Quartic Anharmonic Oscillator
Perturbation Theory in Operator Form
Illustrative Example: The Anharmonic Oscillator
According to our philosophy of choosing simple examples to illustrate the application of the perturbation approaches discussed in this book, in what follows we consider the dimensionless anharmonic oscillator
, (3.195) whereDˆ = dx d The use of boson operators greatly facilitates the calculation becauseaˆandaˆ † are eigenvectors ofH'ˆ0 Substituting x= 1
Solving equation (3.188) forWˆ j andKˆ j+ 1is remarkably simple becausefˆ j is a polynomial function
Table 3.9 Perturbation Theory in Operator Form for the Anharmonic
−3825aˆ † aˆ− 30885 128 of the boson operators and, consequently, a linear combination of eigenvectors ofH'ˆ0 Taking into account that
In the context of operator methods, we establish that H'0(ˆa†)m aˆn = (m−n)(ˆa†)m aˆn and H'0^−1(ˆa†)m aˆn = (m−n)^−1(ˆa†)m aˆn when m equals n This indicates the necessity of selecting Kˆj+1 to eliminate all diagonal terms (ˆa†)m aˆm from fˆj It is crucial to maintain the correct order of noncommuting operators to simplify calculations Therefore, we adopt a normal ordering convention, positioning the creation operator ˆa† to the left of the annihilation operator a, as illustrated in the example.
Table 3.9 presents the initial operators Wˆ j and Kˆ j + 1 for the anharmonic oscillator with M=2 It is important to note that each operator Wˆ j is antihermitian, consistent with the antihermitian nature of Wˆ In one-dimensional models, which lack degeneracy, the eigenvectors |n> of Hˆ0 also serve as eigenvectors of Kˆ By utilizing the relationships [49] ˆ a|n> = √n |n−1> and aˆ † |n> = √(n+1) |n+1>, we can easily compute the perturbation corrections based on the harmonic oscillator quantum number n, as outlined in equations (3.192) and (3.193) For instance, the energy coefficients E n,j are derived from these calculations.
Perturbation theory in operator form can take various equivalent forms, making it a versatile approach in quantum mechanics While it may not be the most practical method for simpler models, such as the one-dimensional anharmonic oscillator, it remains a favored approximate technique for addressing several significant physical problems Comparisons with other methods, like those of Swenson and Danforth, highlight its limitations in certain contexts, yet its utility in more complex scenarios is well-established.
Simple Atomic and Molecular Systems
Introduction
This chapter applies perturbation methods to simple atomic and molecular systems, including the Stark and Zeeman effects in hydrogen and the hydrogen molecular ion in the Born-Oppenheimer approximation The Stark effect in hydrogen is examined in both parabolic and spherical coordinates, allowing for a comparison of the method of Swenson and Danforth and the moment method By applying these methods to a nonseparable problem in spherical coordinates, readers can gain insight into the effectiveness of different approaches in solving complex quantum mechanical problems.
Currently, no coordinate system has been identified that allows the Schrödinger equation for the Zeeman effect in hydrogen to be separable In Chapter 2, we discussed the ground state of this issue using Dalgarno and Stewart's method alongside logarithmic perturbation theory In this section, we extend our analysis to excited states using the moment method, offering valuable additional examples for both nondegenerate and degenerate states.
The hydrogen molecular ion can be separated in elliptical coordinates within the Born-Oppenheimer approximation, but an alternative approach involves expressing the Schrödinger equation in spherical coordinates This method utilizes the moment method for nonseparable problems to derive the electronic energies at large internuclear distances Notably, this example diverges from previous cases due to the perturbation not being a polynomial function of the coordinates, necessitating a Taylor series expansion to apply the moment method effectively.
The Stark Effect in Hydrogen
Parabolic Coordinates
The Hamiltonian operator for the nonrelativistic isolated hydrogen atom in the coordinate repre- sentation is
2m∇ 2 −e 2 r , (4.1) wheremis the reduced mass,ris the distance between the nucleus and the electron, ande >0 and
−eare, respectively, the nucleus and electron charges Accordingly, the dipole moment of the atom isd= −e r If the electric field is directed along thezaxisF=F k, then the interaction energy is
H = −d F=eF z Choosingγ = ¯h 2 /(me 2 )ande 2 /γ =me 4 /h¯ 2 as units of length and energy, respectively, andλ = F γ 2 /e = mγ 3 F e/h¯ 2 as a dimensionless perturbation parameter, then the dimensionless Hamiltonian operatorHˆ = ˆH 0+ ˆH reads
As said before, the Schrửdinger equation for this model is separable in parabolic coordinatesξ r−z≥0, η=r+z≥0,0≤φ=tan(y/x) Whenj =0 this equation reduces to
4 U 1=0, (4.14) where we have chosen the intermediate normalization conditionU 0=1 Allowing bothEandCto depend onλ, the Hellmann–Feynman theorem takes the form
U j,i λ i (4.16) into the equations given above to derive
The calculation of energy coefficients mirrors the approach used in Chapter 3 for the perturbed Coulomb problem, demonstrating that E_j is linear with respect to C_j and exhibits a nonlinear relationship with coefficients C_i for i < j.
To derive the energy coefficients \( E^I_j \) and \( E^{II}_j \), we start with the equation \( C_j = \delta_{j0} - A_j \) and set \( \sigma = -1 \) By solving the equation \( E^I_j = E^{II}_j \) for \( j = 1, 2, \ldots \), we can find the values of \( A_j \) Substituting these results back into either \( E^I_j \) or \( E^{II}_j \) allows for a straightforward calculation, as the equations \( E^I_j - E^{II}_j = 0 \) for \( j > 0 \) are linear with respect to the unknown \( A_j \) Ultimately, this process yields both \( A_j \) and \( E_j \) expressed in terms of the quantum numbers \( n_1, n_2, \) and \( m \) In the program section, we provide simple Maple procedures to facilitate this calculation.
Table 4.1 presents the initial coefficients A_j and E_j related to quantum numbers, with results expressed as n = k_1 + k_2 = n_1 + n_2 + |m| + 1 and q = k_1 - k_2 = n_1 - n_2, simplifying the expressions Notably, the most comprehensive calculation of analytical perturbation corrections to the Stark effect in hydrogen has been conducted using a Maple program that employs an algorithm grounded in algebraic methods.
Spherical Coordinates
Before examining the specific case of the Stark effect in spherical coordinates, it is essential to apply the moment method to a hydrogen atom under a broader range of perturbations.
Table 4.1 Method of Swenson and Danforth for the Stark Effect in Hydrogen in Parabolic Coordinates
By means of the method developed in Appendix A we easily obtain the Laplacian in spherical coordinates
To establish a suitable recurrence relation for the moments of the eigenfunctions of the operator \( \hat{H} \), we define the functions \( f_{i,j,k,m} = \sin(\theta)^i \cos(\theta)^j r^k e^{-\alpha r + i m \phi} \), where \( i, j, k = 0, 1, \ldots \) and \( m = 0, \pm 1, \ldots \) Here, \( i \) and \( j \) represent non-negative integers, while \( m \) can take on both positive and negative integer values, with \( i \) denoting the imaginary unit.
It is not difficult to verify that
2 f i,j− 2 ,k− 2 ,m −.Ef i,j,k,m +λHˆ f i,j,k,m , (4.27) where.E=E+α 2 /2 In order to obtain this equation we have systematically rewritten expressions of the form sin(θ) i− 2 cos(θ) j+ 2 as[sin(θ) i− 2 −sin(θ) i ]cos(θ) j
The application of the moment method to this problem is straightforward if we can writeHˆ as a polynomial function ofrand trigonometric functions ofθandφ For simplicity, here we assume that
The operator Hˆ is independent of the variable φ, indicating that Lˆ z remains a constant of motion, making m a valid quantum number This independence is evident as the subscript m remains unchanged in the recurrence relation (4.27), allowing it to be omitted.
We choose α= 1 n, n=1,2, (4.28) that makes the second term on the right-hand side of equation (4.27) vanish when k = n−1 simplifying the problem Therefore,
2n 2 (4.29) is the energy shift with respect to the energy of the isolated hydrogen atomE 0= −1/(2n 2 ). The recurrence relation (4.27) for the Stark effect becomes
Notice that the subscriptichanges in only one term that vanishes wheni= |m|, and in that case the moments of the eigenfunction
The present moment method restricts the simultaneous treatment of all Stark states due to the non-separability of the Schrödinger equation in spherical coordinates However, we can categorize states based on the relationship between |m| and n To facilitate this, we define k as |m| + 1 + t, where t can take values of -1, 0, or 1, among others.
G j,t =F j,k− 1 (4.33) so that the recurrence relation (4.32) becomes
Expanding the energy and the new moments in Taylor series aboutλ=0
G j,t,p λ p (4.35) we obtain the master recurrence relation
(4.36) valid for all the states discussed below.
We first consider states with|m| =n−1 (k=n+t) Whenj =0 andt = −1 equation (4.34) reduces to−.EG 0 , 0+λG 1 , 1 =0; therefore, if we choose the arbitrary normalization condition
G 0 , 0=1, then we obtain a suitable expression for the energy:.E=λG 1 , 1 We easily calculate all the perturbation corrections to the energy and moments by means of equation (4.36) supplemented with
G 0 , 0 ,q =δ 0 q , E q =G 1 , 1 ,q − 1 (4.37) that come from the normalization condition and from the energy equation, respectively The calcu- lation ofE p requires the moment coefficientsG j,t,q withq=0,1, , p−1,j =0,1, , p−q, t =0,1, ,2(p−q)−1.
We next consider the states with|m| =n−2 Settingj =1 andt=0 we obtain.EG 1 , 1=λG 2 , 2 that suggests the intermediate normalization conditionG 1 , 1=1 leading to the simple energy equation
E=λG 2 , 2 When (j =0, t= −1) and (j =0, t=0) we obtain two equations
1 nG 0 ,− 1+.EG 0 , 0−λ=0, (n−1)G 0 ,− 1+.EG 0 , 1−λG 1 , 2=0 (4.38) which lead to n(n−1)G 0 , 0−G 0 , 1
G 1 , 2−n(n−1) λ=0 (4.39) after elimination of the momentG 0 ,− 1between them Substitution of the expression for the energy into equation (4.39) yields the secular equation n(n−1)G 0 , 0−G 0 , 1
G 2 , 2+G 1 , 2−n(n−1)=0 (4.40) Another useful equation arises whenj =1 andt=1:
The derived equations facilitate the computation of perturbation corrections to energy and moments for states where |m| = n - 2 By expanding the energy (E) and moments (G j,t) in Taylor series around λ = 0, we can obtain the necessary results.
G 2 , 2 ,p−q +G 1 , 2 ,p −n(n−1)δ 0 p =0, (4.43) and the master equation (4.36) withi= |m| =n−2 It is not difficult to verify that the perturbation equations (4.36) and (4.42) leave undetermined only the moment coefficientsG 0 , 0 ,p ,p=0,1,
We obtain them from equation (4.43) which is quadratic inG 0 , 0 , 0whenp=0, and linear inG 0 , 0 ,p for allp >0 Forp=0 we have
Each sign corresponds to one of the two Stark states arising from degenerate unperturbed hydrogenic states In order to obtainE p we have to calculateG j,t,q for allq =0,1, , p−1,j =0,1, , p− q+1 andt =1,2, ,2(p−q).
Table 4.2 presents the energy coefficients for two Stark states derived from degenerate unperturbed states with |m| = n−2, using σ = ±1 to differentiate between them The article does not include the Maple procedures utilized to generate these results, encouraging readers to create their own based on similar programs in the relevant section Notably, the energy coefficients in Table 4.2 align with those obtained from a previous moment method application [58] Additionally, by setting (q = 0, |m| = n−1) and (q = σ, |m| = n−2), the energy coefficients in Table 4.1 correspond to those in Table 4.2.
The projection of angular momentum along the field direction remains a constant of motion, making the quantum number m = 0, ±1, ±2, significant Additionally, the moment method introduces another quantum number σ = ±1 for labeling Stark state pairs In parabolic coordinates, three quantum numbers are evident: m, n₁, and n₂, or alternatively m, n, and q, with q equating to σ in the moment method analysis In spherical coordinates, Stark states are typically identified using the quantum numbers of the isolated hydrogen atom: n = 1, 2, , l = 0, 1, , n-1, and m = 0, ±1, ±2, , ±l, which is applicable at low field strengths The first scenario shows |m| = n - 1 = l, correlating Stark states to hydrogenic states like 1s, 2p ± 1, and 3d ± 2 In contrast, the second scenario reveals |m| = n - 2, where the angular momentum quantum numbers l = n - 2 and n - 1 indicate perturbations that couple hydrogenic state pairs such as (2s, 2p₀) and (3p ± 1, 3d ± 1).
Table 4.2 Moment Method for the Stark Effect in Hydrogen in Spherical Coordinates
The Zeeman Effect in Hydrogen
We have briefly discussed the Zeeman effect in hydrogen inChapter 2to illustrate the application of the method of Dalgarno and Stewart and logarithmic perturbation theory to a nonseparable problem.
It was shown that the relevant part of the dimensionless Hamiltonian operator reads
Therefore, arguing as in the preceding section we obtain the recurrence relation
The equation 2f(i,j) - 2f(k-2,m) - Ef(i,j,k,m) + λf(i+2,j,k+2,m) illustrates that the subscript j remains constant when j(j-1) = 0 This leads to the formation of two distinct sets of states corresponding to the values of j, specifically j = 0 or j = 1, which function as a quantum number Regardless of the selection, the moments are consistently defined.
In order to derive a moment recurrence relation that applies not only to individual states but also to whole classes of them we definei= |m| +2s,k= |m| +j+1+t,s=0,1, ,t = −1,0,1, , and
Expanding the new moments and the energy in Taylor series aboutλ=0 we obtain a master equation
(4.51) that applies to all the cases studied here.
We first consider states with j = 0 The simplest case is|m| = n−1 because three terms of equation (4.50) vanish when s = 0 and t = −1 giving a single expression for the energy
EG 0 , 0=λG 1 , 2that suggests the intermediate normalization conditionG 0 , 0=1 We thus obtain two additional expressions,
To enhance the understanding of the master equation (4.51), we introduce the relationship \( G_{0,0,q} = \delta_{q,0} \) and \( E_q = G_{1,2,q-1} \) To derive \( E_p \), we compute all moment coefficients \( G_{s,t,q} \) for \( q = 0, 1, \ldots, p-1 \), \( s = 0, 1, \ldots, p-q \), and \( t = 0, 1, \ldots, 3(p-q)-1 \) When \( |m| = n-2 \) (where \( j = 0 \)), it is impossible to nullify three terms in the recurrence relation (4.50) simultaneously By selecting \( (s,t) = (0,-1) \) and \( (s,t) = (0,0) \), we derive two equations: \( (n-1)G_{0,-1} + E G_{0,0} - \lambda G_{1,2} = 0 \) and \( (n-1)G_{0,-1} + E G_{0,1} - \lambda G_{1,3} = 0 \) Eliminating \( G_{0,-1} \) from these equations yields a valuable expression for energy: \([n(n-1)G_{0,0} - G_{0,1}]E - [n(n-1)G_{1,2} - G_{1,3}]\lambda = 0\) The normalization condition \( n(n-1)G_{0,0} - G_{0,1} = 1 \) simplifies our formula to \( E = \lambda[n(n-1)G_{1,2} - G_{1,3}] \) Finally, when \( s = 0 \) and \( t = 1 \), equation (4.50) simplifies to \( nG_{0,0} + (n-1)E G_{0,2} - \lambda G_{1,4} = 0 \), allowing us to solve for \( G_{0,0} \).
Expanding the energy and moments in Taylor series aboutλ=0 we obtain
E p =n(n−1)G 1 , 2 ,p− 1−G 1 , 3 ,p− 1 (4.58) in addition to the master equation (4.51) In order to obtain the energy coefficientE p we needG s,t,q withq =0,1, , p−1,s=0,1, , p−q, andt =1,2, ,3(p−q).
The states mentioned can be classified as nondegenerate since the perturbation does not couple them However, a different scenario arises when |m| = n - 3 Utilizing the general recurrence relation for the moments, we derive the following equations.
(n−2)G 0 ,− 1+1 nG 0 , 0+.EG 0 , 1−λG 1 , 3 = 0 (4.62) when(s, t)is respectively equal to(0,−1),(0,1),(1,1), and(0,0) It is left to the reader to derive the following working expressions
The equation \(2 G_{1, 2} - G_{1, 4}(2n-3) + G_{1, 3} = 0\) serves as an arbitrary normalization condition By expanding these equations in a Taylor series around \(\lambda = 0\) and applying the master equation (4.51) with suitable values of \(|m|\) and \(j\), we can derive all energy and moment coefficients To calculate \(E_p\), we require \(G_{s,t,q}\) with \(q = 0, 1, \ldots, p-1\), \(s = 0, 1, \ldots, p-q+1\), and \(t = 2, 3, \ldots, 3(p-q)+1\) The coefficient of order \(q\) in the expansion of equation (4.66) is linear in the moment coefficient \(G_{0, 1, q}\) for \(q > 0\) and quadratic in \(G_{0, 1, 0}\) when \(q = 0\) This latter scenario leads to the secular equation for the two degenerate states coupled by the perturbation, yielding two roots.
In our analysis of states with j = 1, we find that the simplest scenario occurs when |m| = n - 2, as attempts to derive appropriate equations for |m| = n - 1 prove unsuccessful By applying the normalization condition G(0, 0) = 1, we arrive at the energy expression E = λG(1, 2).
In order to obtainE p we have to calculateG s,t,q forq =0,1, , p−1,s=0,1, , p−q, and t =0,1, ,3(p−q)−1.
The case |m| = n - 3 presents no significant challenges and is thus left for the reader to explore independently Nevertheless, for thoroughness, we will present the key equations that follow All energy and moment coefficients are derived from the master equation (4.51) along with supplementary expressions.
E p =n(n−1)G 1 , 2 ,p− 1−G 1 , 3 ,p− 1 , (4.71) where equation (4.70) is an arbitrary normalization condition To obtainE p we have to calculate the moment coefficientsG s,t,q withq =0,1, , p−1,s=0,1, , p−q, andt =1,2, ,3(p−q).
Table 4.3 Moment Method for the Zeeman Effect in Hydrogen(Continued)
Table 4.3 (Cont.)Moment Method for the Zeeman Effect in Hydrogen(Continued)
Table 4.3 (Cont.)Moment Method for the Zeeman Effect in Hydrogen
Table 4.3 shows the first energy coefficients for all the states considered above in terms of the principal quantum numbernandR= ±√
16n 2 −48n+41 Analytical expressions of greater order are much longer and, most probably, of no use for the reader.
With adequate computer memory, users can efficiently calculate additional analytic energy coefficients beyond those listed in Table 4.3 using simple Maple procedures The program section demonstrates the most complex scenario (j = 0, |m| = n−3), but readers can easily adapt the main procedure to explore other cases Additionally, setting the principal quantum number n to 1, 2, etc., for a specific state significantly enhances calculation speed and reduces memory usage.
Numerous authors have calculated energy coefficients for the Zeeman effect in hydrogen, yet their findings show significant discrepancies, as highlighted in a previous application of the moment method We assert that the energy coefficients presented in Table 4.3, which align with those in reference [59], are accurate.
The moment method simplifies the calculation of energy coefficients compared to the Dalgarno and Stewart method and logarithmic perturbation theory, particularly in handling excited states An effective alternative is the expansion of the perturbed state in a basis set of unperturbed states, which allows for a systematic algebraic calculation of necessary matrix elements This algebraic approach is advantageous for calculating system properties beyond energy, while the moment method offers simpler programming solutions Additionally, the classification and labeling of states within the moment method do not explicitly rely on the known properties of Zeeman states, as the model Hamiltonian remains invariant under specific substitutions The chosen functions for constructing moments exhibit symmetry characteristics that align with these invariances.
To derive a nontrivial recurrence relation for the moments, the symmetry of the functions f i,j,k,m (r, θ, φ), dictated by the values of j and |m|, must align with the symmetry of the selected Zeeman state.
The unperturbed eigenfunctions are radial factors times the spherical harmonicsY l,m (θ, φ)that satisfyY l,m (−θ, φ) = (−1) |m| Y l,m (θ, φ)andY l,m (θ+π, φ)= (−1) l Y l,m (θ, φ)[40] Therefore, whenλ=0 we expect thatl= |m| +j+2u,u=0,1, For the class of states withj =0 and
In the context of Zeeman states derived from hydrogenic configurations, we observe that when |m| = n−1, the corresponding angular momentum quantum number l equals |m|, leading to states such as 1s, 2p ± 1, and 3d ± 2 For j = 0 and |m| = n−2, we again find l = |m|, resulting in Zeeman states from 2s, 3p ± 1, and 4d ± 2 The case of j = 0 and |m| = n−3 introduces two scenarios: l = |m| and l = |m| + 2, yielding pairs of hydrogenic states like (3s, 3d 0) and (4p ± 1, 4f ± 1) that are coupled by perturbation When j = 1 and |m| = n−2, the unperturbed states include 2p 0, 3d ± 1, and 4f ± 2 Lastly, for j = 1 and |m| = n−3, the relevant states are 3p 0, 4d ± 1, and 5f ± 2.
Applying the moment method to a broader range of perturbations is straightforward, as demonstrated by previous studies, such as the analysis of the hydrogen atom subjected to parallel electric and magnetic fields.