We can represent a state as a probability amplitude inx-space or ink-space, and we can switch between both representations through Fourier transformation. The state itself is apparently independent from which representation we choose, just like a vector is independent from the particular basis in which we expand the vector. In Chap.7we will derive a wave functionψ1s(x, t )for the relative motion of the proton and the electron in the lowest energy state of a hydrogen atom.
However, it does not matter whether we use the wave function ψ1s(x, t ) in x- space or the Fourier transformed wave functionψ1s(k, t )ink-space to calculate observables for the ground state of the hydrogen atom. Every information on the state can be retrieved from each of the two wave functions. We can also contemplate more exotic possibilities like writing theψ1s state as a linear combination of the oscillator eigenstates that we will encounter in Chap.6. There are infinitely many possibilities to write down wave functions for one and the same quantum state, and all possibilities are equivalent. Therefore wave functions are only particular representations of a state, just like the componentsai|vof a vector|vin anN- dimensional vector space provide only a representation of the vector with respect to a particular basis|ai, 1≤i≤N.
This motivates the following adaptation of bra-ket notation: The (generically time-dependent) state of a quantum system is|ψ (t ), and thex-representation is just the specification of|ψ (t )in terms of its projection on a particular basis,
ψ(x, t )= x|ψ (t ), (4.55) where the “basis” is given by the non-enumerable set of “x-eigenkets”:
4.2 Bra-ket Notation in Quantum Mechanics 75
x|x =x|x. (4.56)
Herexis the operator, or rather a vector of operatorsx=(x,y,z), andx=(x, y, z) is the corresponding vector of eigenvalues.
In advanced quantum mechanics, the operators for location or momentum of a particle and their eigenvalues are sometimes not explicitly distinguished in notation, but for the experienced reader it is always clear from the context whether e.g.x refers to the operator or the eigenvalue. We will denote the operatorsxandpfor location and momentum and their Cartesian components with upright notation,x= (x,y,z),p=(px,py,pz), while their eigenvalue vectors and Cartesian eigenvalues are written in italics notation,x=(x, y, z)andp= ¯hk=(px, py, pz). However, this becomes very clumsy for non-Cartesian components of the operatorsxandp, but once we are at the stage where we have to use e.g. both location operators and their eigenvalues in polar coordinates, you will have so much practice with bra-ket notation that you will infer from the context whether e.g.rrefers to the operatorr = x2+y2+z2or to the eigenvaluer =
x2+y2+z2. Some physical quantities have different symbols for the related operator and its eigenvalues, e.g.H for the energy operator andEfor its eigenvalues,
H|E =E|E, (4.57)
so that in these cases the use of standard italics mathematical notation for the operators and the eigenvalues cannot cause confusion.
Expectation values of observables are often written in terms of the operator or the observable, e.g.x ≡ x,E ≡ H etc., but explicit matrix elements of operators should always explicitly use the operator, e.g.ψ|x|ψ,ψ|H|ψ.
The “momentum-eigenkets” provide another basis of quantum states of a particle,
p|k = ¯hk|k, (4.58)
and the change of basis looks like the corresponding equation in linear algebra: If we have two sets of basis vectors |ai,|ba, then the components of a vector|v with respect to the new basis|baare related to the|ai-components via (just insert
|v = |aiai|v)
ba|v = ba|aiai|v, (4.59) i.e. the transformation matrixTai = ba|aiis just given by the components of the old basis vectors in the new basis.
The corresponding equation in quantum mechanics for the|xand|kbases is x|ψ (t ) =
d3kx|kk|ψ (t ) = 1
√2π3
d3kexp(ikãx)k|ψ (t ), (4.60)
which tells us that the expansion coefficients of the vectors|kwith respect to the
|x-basis are just
x|k = 1
√2π3
exp(ikãx). (4.61)
The Fourier decomposition of theδ-function implies that these bases are self-dual, e.g.
x|x =
d3kx|kk|x = 1 (2π )3
d3kexp[ikã(x−x)] =δ(x−x).
The scalar product of two states can be written in terms of|x-components or|k- components
ϕ(t )|ψ (t ) =
d3xϕ(t )|xx|ψ (t ) =
d3xϕ+(x, t )ψ(x, t )
=
d3xϕ(t )|kk|ψ (t ) =
d3xϕ+(k, t )ψ(k, t ). (4.62) To get some practice with bra-ket notation let us derive the x-representation of the momentum operator. We know Eq. (4.58) and we want to find out what the x-components of the state p|ψ (t ) are. We can accomplish this by inserting the decomposition
|ψ (t ) =
d3k|kk|ψ (t ) (4.63) intox|p|ψ (t ),
x|p|ψ (t ) =
d3kx|p|kk|ψ (t ) =
d3khk¯ x|kk|ψ (t ). (4.64) However, Eq. (4.61) implies
hkx|k = ¯¯ h
i∇x|k, (4.65)
and substitution into Eq. (4.64) yields x|p|ψ (t ) = ¯h
i∇
d3kx|kk|ψ (t ) = ¯h
i∇x|ψ (t ). (4.66) This equation yields in particular the matrix elements of the momentum operator in the|x-basis,
4.2 Bra-ket Notation in Quantum Mechanics 77
x|p|x = ¯h
i∇δ(x−x). (4.67)
Equation (4.66) means that thex-expansion coefficientsx|p|ψ (t )of the new state p|ψ (t )can be calculated from the expansion coefficientsx|ψ (t )of the old state
|ψ (t )through application of−ih¯∇. In sloppy terminology this is the statement “the x-representation of the momentum operator is−ih¯∇”, but the proper statement is Eq. (4.66),
x|p|ψ (t ) = ¯h
i∇x|ψ (t ). (4.68)
The quantum operatorpacts on the quantum state|ψ (t ), the differential operator
−ih¯∇acts on the expansion coefficientsx|ψ (t )of the state|ψ (t ).
The corresponding statement in linear algebra is that a linear transformationA transforms a vector|vaccording to
|v → |v =A|v, (4.69)
and the transformation in a particular basis reads
ai|v = ai|A|v = ai|A|ajaj|v. (4.70) The operatorAacts on the vector, and its representationai|A|ajin a particular basis acts on the components of the vector in that basis.
Bra-ket notation requires a proper understanding of the distinction between quantum operators (like p) and operators that act on expansion coefficients of quantum states in a particular basis (like −ih¯∇). Bra-ket notation appears in virtually every equation of advanced quantum mechanics and quantum field theory.
It provides in many respects the most useful notation for recognizing the elegance and power of quantum theory.
Equations equivalent to Eqs. (4.56), (4.58), and (4.66) are contained in x=
d3x|xxx| =
d3k|ki ∂
∂kk|, (4.71)
p=
d3k|k¯hkk| =
d3x|x ¯h i
∂
∂xx|. (4.72)
Here we used the very convenient notation∇ ≡ ∂/∂x for the del operator inx space, and∂/∂kfor the del operator inkspace. One often encounters several copies of several vector spaces in an equation, and this notation is extremely useful to distinguish the different del operators in the different vector spaces.
In the previous discussion we have figured out that the substitutionp → −ih¯∇ in x-representation and the corresponding substitution for x in k-representation corresponds to the equations
x|p= ¯h i
∂
∂xx|, k|x=i ∂
∂kk|. (4.73)
However, we can also calculate the derivatives of the eigenkets with respect to their eigenvalues. Equations (4.71) and (4.72) also imply5
∂
∂k|k =ix|k, ih¯ ∂
∂x|x =p|x. (4.74)
Functions of operators are operators again. An important example are the operators V (x)for the potential energy of a particle. The eigenkets ofxare also eigenkets of V (x),
V (x)|x =V (x)|x, (4.75) and the matrix elements inxrepresentation are
x|V (x)|x =V (x)δ(x−x). (4.76) The single-particle Schrửdinger equation (1.63) is in representation free notation
ih¯ d
dt|ψ (t ) =H|ψ (t ) = p2
2m|ψ (t ) +V (x)|ψ (t ). (4.77) We recover thex representation already used in (1.63) through projection on x| and substitution of
1=
d3x|xx|, (4.78)
ih¯ ∂
∂tx|ψ (t ) = − ¯h2
2mx|ψ (t ) +V (x)x|ψ (t ). (4.79) The definition of adjoint operators in representation-free bra-ket notation is
ϕ|A|ψ = ψ|A+|ϕ+. (4.80) This implies in particular that the “bra vector”|adjoint to the “ket vector”| = A|ψsatisfies
| = ψ|A+. (4.81)
5We will see in a little while that these relations can also be derived using shift operators. This is explained in Problem6.6b.