1. Wave function
PROBLEMS
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2. Expectation values
Given a plane wave 𝜓 𝑥 = 𝐴𝑒 𝑖𝑘𝑥 , what is (1) the expectation value for 𝑝, the 𝑥-component of momentum,
and (2) the expectation value of the energy operator?
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3. Stationary states
Show 𝐻2 = 𝐸 2 for separable solutions.
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4. Stationary states
Show that the general solution below is a solution to the time-dependent Schrödinger equation. Hint: Plug the
𝜕
෡
general solution into 𝑖ℏ Ψ = 𝐻Ψ.
𝜕𝑡
∞
Ψ 𝑥, 𝑡 = ෍ 𝑐𝑛 𝜓𝑛 𝑥 𝑒 −𝑖𝐸𝑛 𝑡/ℏ
𝑛=1
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5. Potential well problem
Consider the 𝑛 = 1 eigenfunction, 𝜓1 𝑥 =
2
𝜋𝑥
sin
𝑎
𝑎
, of an infinite potential well. Compute 𝑥 .
Potentially useful information
Neamen
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6. Potential well problem
Potentially useful information
Griffiths
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