Chapter 9: Q49E (page 406)

Verify that the probabilities shown in Table 9.1 for four distinguishable oscillators sharing energy $2 δE$ agree with the exact probabilities given by equation (9-9).

Short Answer

Expert verified

The probabilities agree with the result from equation 1.

$P (n) = \frac{(M - n + N - 2)!}{(M - n)! (N - 2)!} / \frac{(M + N - 1)!}{M! (N - 1)!}$

Step by step solution

Concept of probability of finding a particle at an energy state

Probability of finding a particle at an energy state is:

$P (n) = \frac{(M - n + N - 2)!}{(M - n)! (N - 2)!} / \frac{(M + N - 1)!}{M! (N - 1)!}$ ……. (1)

where $M$ , is the sum of nonnegative distinct quantum numbers $n_{i}$ and $N$ is the total number of particles.

Calculate the probability using equation (1)

Here, $M = 2, N = 4$ , and the allowed values of $n$ are $n = 0, 1, 2$ . Let us first consider $n = 0$ .

The probability is:

role="math" localid="1660022722768" $\begin{array}{l} P (n = 0) = \frac{(2 - 0 + 4 - 2)!}{(2 - 0)! (4 - 2)!} / \frac{(2 + 4 - 1)!}{2! (4 - 1)!} \\ = \frac{4!}{2! 2!} / \frac{5!}{2! 3!} \\ = 6 / 10 \\ = 0 .6 \end{array}$

For $n = 1$ , we get:

$\begin{array}{l} P (n = 1) = \frac{(2 - 1 + 4 - 2)!}{(2 - 1)! (4 - 2)!} / \frac{(2 + 4 - 1)!}{2! (4 - 1)!} \\ = \frac{3!}{1! 2!} / \frac{5!}{2! 3!} \\ = 3 / 10 \\ = 0.3 \end{array}$

Finally, for $n = 2$ , we get:

$\begin{array}{l} P (n = 2) = \frac{(2 - 2 + 4 - 2)!}{(2 - 2)! (4 - 2)!} / \frac{(2 + 4 - 1)!}{2! (4 - 1)!} \\ = \frac{2!}{0! 2!} / \frac{5!}{2! 3!} \\ = 1 / 10 \\ = 0 .1 \end{array}$

We have thus verified that the probabilities agree with the result from equation 1.

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Verify that the probabilities shown in Table 9.1 for four distinguishable oscillators sharing energy $2 δE$ agree with the exact probabilities given by equation (9-9).

Short Answer

Step by step solution

Concept of probability of finding a particle at an energy state

Calculate the probability using equation (1)

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Verify that the probabilities shown in Table 9.1 for four distinguishable oscillators sharing energy 2δEagree with the exact probabilities given by equation (9-9).

Short Answer

Step by step solution

Concept of probability of finding a particle at an energy state

Calculate the probability using equation (1)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Dynamics

Further Mechanics and Thermal Physics

Geometrical and Physical Optics

Medical Physics

Circular Motion and Gravitation

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

Verify that the probabilities shown in Table 9.1 for four distinguishable oscillators sharing energy $2 δE$ agree with the exact probabilities given by equation (9-9).