Physical Chemistry Third Edition

720 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

We now make a set of many measurements ofA, ensuring somehow that the system is

in the same state prior to each measurement. Each outcome will be an eigenvalue of

Â. Let the fraction that results in the valueajbe equal topj. By Eq. (16.4-33),

|cj(prior)|

√

pj (16.6-7)

If all of the measurements give the same result, sayai, and if the state corresponding

tofiis nondegenerate, thenpiequals unity and we can assert that the state prior to the

measurement must have been the state corresponding tofi. If the measurements yield

more than one value, we can determine the magnitudes of thecj(prior) coefficients from

the probabilities of the different eigenvalues. We cannot know the real and imaginary

parts of each expansion coefficient, so we cannot know exactly what the wave function

was prior to the measurements. If the eigenvalues are degenerate, we cannot even tell

the magnitudes of the individual coefficients prior to the measurements, but can only

get a collective value for each energy level.

EXAMPLE16.27

The energy of a harmonic oscillator is measured repeatedly with the oscillator restored to the

same unknown state before each measurement. The results are summarized as follows:

Value of the energy Probability

hν/ 2 0.375

3 hν/ 2 0.25

5 hν/ 2 0.25

7 hν/ 2 0.125

What can you say about the state prior to the measurement?

Solution

The only values obtained correspond toE 0 ,E 1 ,E 2 , andE 3. The wave function just prior to

the measurement must be represented by

ψ(prior)c 0 ψ 0 +c 1 ψ 1 +c 2 ψ 2 +c 3 ψ 3

where

|c 0 |

√

0. 375

|c 1 |

√

0. 25

|c 2 |

√

0. 25

|c 3 |

√

0. 125

We cannot give the real and imaginary parts of theccoefficients.

Exercise 16.16

Show that the wave function represented by the linear combination in the previous example is

normalized.