Physical Chemistry Third Edition

718 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

EXAMPLE16.26

Assume that we want to find the location of an electron in a box of length 1.0 nm to the nearest

0.1 nm (one-tenth of the length of the box). This requires X-rays with wavelength no longer

than 0.1 nm, since the scattering of a wave can reveal the position of the scatterer only to an

uncertainty of one wavelength. Compare the energy of a photon of wavelength 0.10 nm with

E 1 for electron in a one-dimensional box of length 1.0 nm.

Solution

E(photon)hν

hc

λ



(6. 6261 × 10 −^34 J s)(3. 00 × 108 ms−^1 )

1. 00 × 10 −^10 m

 1. 99 × 10 −^15 J

E(electron)

h^2

8 ma^2



(6. 6261 × 10 −^34 Js)^2

8(9. 1094 × 20 −^31 kg)(1. 00 × 10 −^9 m)^2

 6. 02 × 10 −^20 J

The photon energy is about 30,000 times as large as the kinetic energy of the electron, so

that the energy transferred to the electron in a measurement can be much larger than the

original kinetic energy of the electron. We cannot use electromagnetic radiation to determine

the position of an electron without drastically changing its state.

Let us try to find out what we can about the state of the system if we make a

single measurement of a variableAwith outcomeaj, one of the eigenvalues of the

operatorÂ. Prior to the measurement the wave function could have been represented

by a linear combination of all of the eigenfunctions of the operator. From our discussion

of Postulate 4, we know that if the outcome of the measurement is the eigenvalueaj,

then we can say that the coefficient of at least one of the eigenfunctions belonging to

that eigenvalue was nonzero in this linear combination. However, this is all that we can

say about the state prior to the measurement.

Information about the State of a System after

a Measurement

After the measurement the wave function is an eigenfunction belonging to the eigenvalue

that was the result of the measurement. If the eigenvalue is nondegenerate, the mea-

surement has put the system it into the state corresponding to that eigenvalue. If the

outcome of a measurement of a variableAis an eigenvalue that is degenerate, the act of

measurement has put it into a state corresponding to a linear combination of all eigen-

functions that correspond to that eigenvalue. Say that the outcome of the measurement

of the variableAisaiand that this eigenvalue has degeneracygi.

ψ(after)

∑gi

j 1

cjfj (16.6-1)

wheref 1 ,f 2 ,...,fgiare the eigenfunctions ofÂcorresponding to the observed

eigenvalueai. Only the functions with the same value for the eigenvalue are included

in the sum, but we do not know what the coefficientsc 1 ,c 2 ,c 3 ,...,cgiare. We have

gained some information, but we do not yet know exactly what the state is.