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1 = (x) %3D L/2

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: \[ \sum_{s=1}^{\infty} \frac{1}{(2s-1)^2} = \frac{\pi^2}{8}. \]
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

ψ=1L

Total probability is found by integrating from 0 to L

P=∫0Lψ*ψ dxP=∫0L1L1LdxP=1L∫0LdxP=1L(L-0)P=1L×LP=1

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