374 Quantum mechanics
1933). Here ̄his related to Planck’s constant by ̄h=h/ 2 πand is also referred to as
Planck’s constant. Since eqn. (9.2.32) is a first-order differential equation, it must be
solved subject to an initial condition|Ψ(0)〉. Interestingly, eqn. (9.2.32) bears a marked
mathematical similarity to the classical equation that determines the evolution of the
phase space vector ̇x =iLx. The Schr ̈odinger equation can be formally solved to yield
the evolution
|Ψ(t)〉= e−i
Hˆt/ ̄h
|Ψ(0)〉. (9.2.33)
Again, note the formal similarity to the classical relation x(t) = exp(iLt)x(0). The
unitary operator
Uˆ(t) = e−iHˆt/ ̄h (9.2.34)is known as thetime evolution operatoror thequantum propagator. The termunitary
means thatUˆ†(t)Uˆ(t) =Iˆ. Consequently, the action ofUˆ(t) on the state vector cannot
change the magnitude of the vector, only its direction. This is crucial, as|Ψ(t)〉must
always be normalized to 1 in order that it generate proper probabilities. Suppose the
eigenvectors|Ek〉and energy levels or eigenvaluesEkof the Hamiltonian are known
from the eigenvalue equation
Hˆ|Ek〉=Ek|Ek〉. (9.2.35)It is then straightforward to show that
|Ψ(0)〉=∑
k|Ek〉〈Ek|Ψ(0)〉|Ψ(t)〉=∑
ke−iEkt/ ̄h|Ek〉〈Ek|Ψ(0)〉. (9.2.36)If we know the initial amplitudes for obtaining the various eigenvaluesofHˆ in an
experiment designed to measure the energy, the time evolution of the state vector can
be determined. In general, the calculation of the eigenvectors andeigenvalues ofHˆ is
an extremely difficult problem that can only be solved for systems witha very small
number of degrees of freedom, and alternative methods for calculating observables are
typically needed.
9.2.5 Position and momentum operators
Up to now, we have formulated the theory of measurement in quantum mechanics
for observables with discrete eigenvalue spectra. While there certainly are observables
that satisfy this condition, we must also consider operators whosespectra are possibly
continuous. The most notable examples are the position and momentum operators
corresponding to the classical position and momentum variables.^3 In infinite space,
the classical position and momentum variables are continuous, so that in a quantum
description, we require operators with continuous eigenvalue spectra. If ˆxand ˆpdenote
(^3) Note, however, that there are important cases in which the momentum eigenvalues are discrete.
An example is a free particle confined to a finite spatial domain, where the discrete momentum
eigenvalues are related to the properties of standing waves. This case will be discussed in Section 9.3.