The statistical interpretation of the wave function ψ(x, t ) implies that the wave functions of single stable particles should be normalized,
d3x|ψ(x, t )|2=1. (2.33) Time-dependence plays no role and will be suppressed in the following investiga- tions.
2.2 Self-Adjoint Operators and Completeness of Eigenstates 31
Fig. 2.1 Comparison of 1/xwith the weight factorK(x)
Indeed, we have to require a little more than just normalizability of the wave function ψ(x) itself, because the functions ∇ψ(x), ψ(x), and V (x)ψ(x) for admissible potentialsV (x)should also be square integrable. We will therefore also encounter functionsf (x)which may not be normalized, although they are square integrable,
d3x|f (x)|2<∞. (2.34) Letψ(x)andφ (x)be two square integrable functions. The identity
d3x|ψ(x)−λφ (x)|2≥0 (2.35) yields with the choice
λ=
d3xφ+(x)ψ(x)
d3x|φ (x)|2 (2.36)
the Schwarz inequality
d3xφ+(x)ψ(x) 2≤
d3x|ψ(x)|2
d3xφ (x)2. (2.37) The differential operators −ih¯∇ and −(h¯2/2m), which we associated with momentum and kinetic energy, and the potential energyV (x)all have the following properties,
d3xφ+(x)h¯
i∇ψ(x)=
d3xψ+(x)h¯ i∇φ (x)
+
, (2.38)
d3xφ+(x)ψ(x)=
d3xψ+(x)φ (x) +
, (2.39)
and
d3xφ+(x)V (x)ψ(x)=
d3xψ+(x)V (x)φ (x) +
. (2.40)
Eq. (2.40) is a consequence of the fact thatV (x)is a real function. Equations (2.38) and (2.39) are a direct consequence of partial integrations and the fact that boundary terms at|x| → ∞vanish under the assumptions that we had imposed on the wave functions.
If two operatorsAxandBx have the property
d3xφ+(x)Axψ(x)=
d3xψ+(x)Bxφ (x) +
, (2.41)
forallwave functions of interest, thenBxis denoted asadjointto the operatorAx. The mathematical notation for the adjoint operator toAxisA+x,
Bx=A+x. (2.42)
Complex conjugation of (2.41) then immediately tells usBx+=Ax.
An operator with the propertyA+x =Axis denoted as aself-adjointorhermitian operator.2Self-adjoint operators are important in quantum mechanics because they yield real expectation values,
2We are not addressing matters of definition of domains of operators in function spaces, see e.g.
[94] or Problem2.6. If the operatorsA+x andAx can be defined on different classes of functions, and A+x = Ax holds on the intersections of their domains, thenAx is usually denoted as a symmetric operator. The notion of self-adjoint operator requires identical domains for bothAx andA+x such that the domain of neither operator can be extended. If the conditions on the domains are violated, we can e.g. have a situation whereAx has no eigenfunctions at all, or where the eigenvalues of Ax are complex and the set of eigenfunctions is overcomplete. Hermiticity is
2.2 Self-Adjoint Operators and Completeness of Eigenstates 33
(Aψ)+=
d3xψ+(x)Axψ(x) +
=
d3xψ+(x)A+xψ(x)
=
d3xψ+(x)Axψ(x)= Aψ. (2.43)
Observable quantities like energy or momentum or location of a particle are there- fore implemented through self-adjoint operators, e.g. momentumpis implemented through the self-adjoint differential operator−ih¯∇. We have seen one method to figure this out in equation (1.73). We will see another method in Eqs. (4.64) and (4.66).
Self-adjoint operators have the further important property that their eigenfunc- tions yield complete sets of functions. Schematically this means the following:
Suppose we can enumerate all constantsanand functionsψn(x)which satisfy the equation
Axψn(x)=anψn(x) (2.44)
with the set of discrete indicesn. The constantsanareeigenvaluesand the functions ψn(x)areeigenfunctionsof the operatorAx. Hermiticity of the operatorAximplies orthogonality of eigenfunctions for different eigenvalues,
an
d3xψm+(x)ψn(x)=
d3xψm+(x)Axψn(x)
=
d3xψn+(x)Axψm(x) +
=am
d3xψm+(x)ψn(x) (2.45) and therefore
d3xψm+(x)ψn(x)=0 if an=am. (2.46) However, even ifan = am for different indicesn = m (i.e. if the eigenvaluean isdegeneratebecause there exist at least two eigenfunctions with the same eigen- value), one can always chose orthonormal sets of eigenfunctions for a degenerate eigenvalue. We therefore require
d3xψm+(x)ψn(x)=δm,n. (2.47) Completeness of the set of functionsψn(x)means that an “arbitrary” functionf (x) can be expanded in terms of the eigenfunctions of the self-adjoint operatorAx in the form
sometimes defined as equivalent to symmetry or as equivalent to the more restrictive notion of self-adjointness of operators. We define Hermiticity as self-adjointness.
f (x)=
n
cnψn(x) (2.48)
with expansion coefficients cn=
d3xψn+(x)f (x). (2.49) If we substitute Eq. (2.49) into (2.48) and (in)formally exchange integration and summation, we can express the completeness property of the set of functionsψn(x) in thecompleteness relation
n
ψn(x)ψn+(x)=δ(x−x). (2.50) Both the existence and the meaning of the series expansions (2.48) and (2.49) depends on what large a class of “arbitrary” functions f (x) one considers.
Minimal constraints require boundedness of f (x), and continuity if the series (2.48) is supposed to converge pointwise. The default constraints in non-relativistic quantum mechanics are continuity of wave functions ψ(x)to ensure validity of the Schrửdinger equation with at most finite discontinuities in potentials V (x), and normalizability. Under these circumstances the expansion (2.48) and (2.49) for a wave function f (x) ≡ ψ(x) will converge pointwise to ψ(x). However, it is convenient for many applications of quantum mechanics to use limiting forms of wave functions which are not normalizable in the sense of Eq. (2.33) any more, e.g. plane wave statesψk(x) ∝ exp(ikã x), and we will frequently also have to expand non-continuous functions, e.g. functions of the formf (x) = V (x)ψ(x)with a discontinuous potentialV (x). However, finally we only have to use expansions of the form (2.48) and (2.49) in the evaluation of integrals of the form
d3xg+(x)f (x), and here the concept ofconvergence in the meancomes to our rescue in the sense that substitution of the series expansion (2.48) and (2.49) in the integral will converge to the same value of the integral, even if the expansion (2.48) and (2.49) does not converge pointwise to the functionf (x).
A more thorough discussion of completeness of sets of eigenfunctions of self- adjoint operators in the relatively simple setting of wave functions confined to a finite one-dimensional interval is presented in Appendix C. However, for a first reading I would recommend to accept the series expansions (2.48) and (2.49) with the assurance that substitutions of these series expansions is permissible in the calculation of observables in quantum mechanics.