1 = (x) %3D L/2
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![**Problem 1.** Consider a particle in an infinite 1D potential well of width \( L \). Calculate the probabilities \( w_n \) of measurement of different energies \( E_n \) of a particle which at \( t = 0 \) has a wave function independent of coordinate:
\[
\psi(x) = \frac{1}{L^{1/2}}.
\]
Show that the sum of all probabilities equals 1:
\[
\sum_{n=1}^{\infty} w_n = 1.
\]
**HINT:** You may need the sum:
\[
\sum_{s=1}^{\infty} \frac{1}{(2s-1)^2} = \frac{\pi^2}{8}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85e4b871-9d1c-4ae2-b799-fb57954f3d49%2F0fc1d14e-ce50-49db-a3a3-00dd259487e3%2Fpo0x3fk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1.** Consider a particle in an infinite 1D potential well of width \( L \). Calculate the probabilities \( w_n \) of measurement of different energies \( E_n \) of a particle which at \( t = 0 \) has a wave function independent of coordinate:
\[
\psi(x) = \frac{1}{L^{1/2}}.
\]
Show that the sum of all probabilities equals 1:
\[
\sum_{n=1}^{\infty} w_n = 1.
\]
**HINT:** You may need the sum:
\[
\sum_{s=1}^{\infty} \frac{1}{(2s-1)^2} = \frac{\pi^2}{8}.
\]
Expert Solution
Step 1
Total probability is found by integrating from 0 to L
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