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3. A particle in an infinite square well at t=0 has a probability density that is uniform in the left half (0

3. A particle in an infinite square well at t=0 has a probability density that is uniform in the left half (0

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I need some help with this quantum mechanics question.

**Problem Statement:** A particle in an infinite square well at \( t = 0 \) has a probability density that is uniform in the left half (\( 0 < x < L/2 \)), and 0 in the right half. (This is a bit unphysical, because \( \psi(x, t = 0) \neq 0 \) at the left wall of the potential well, but it makes the math easier for this question.) Assume that \( \psi(x, t = 0) \) is a positive, real function. **a)** Sketch \( \psi(x, t = 0) \) and write a mathematical formula for it (e.g. a “piecewise” formula specifying it in the regions \( 0 < x < L/2 \) and \( L/2 < x < L \)). Make sure your formula is normalized! **b)** Find a general formula (valid for all \( n \)) for the probability that a measurement of energy yields \( E_n \), and evaluate this formula for the first 4 cases (\( n = 1 \) to 4). Explain what happens when \( n = 4 \) (Explain the “math” answer using a graph!)
Transcribed Image Text:**Problem Statement:** A particle in an infinite square well at \( t = 0 \) has a probability density that is uniform in the left half (\( 0 < x < L/2 \)), and 0 in the right half. (This is a bit unphysical, because \( \psi(x, t = 0) \neq 0 \) at the left wall of the potential well, but it makes the math easier for this question.) Assume that \( \psi(x, t = 0) \) is a positive, real function. **a)** Sketch \( \psi(x, t = 0) \) and write a mathematical formula for it (e.g. a “piecewise” formula specifying it in the regions \( 0 < x < L/2 \) and \( L/2 < x < L \)). Make sure your formula is normalized! **b)** Find a general formula (valid for all \( n \)) for the probability that a measurement of energy yields \( E_n \), and evaluate this formula for the first 4 cases (\( n = 1 \) to 4). Explain what happens when \( n = 4 \) (Explain the “math” answer using a graph!)
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