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Non-Stationary State

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Exercise

https://texercises.com/exercise/non-stationary-state/

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Exercise:
Derive the probability density for the state psixt fracsqrtpsi_xt+fracsqrtpsi_xt of the infinite potential well. Compare the result to the probability density of a stationary state.

Solution:
The wave function is psixt fracAsqrtleftsink_ xe^iomega_ t+sink_ xe^iomega_ t right It follows for the probability density Pxt psi^*xtpsixt fracA^leftsink_ xe^iomega_ t+sink_ xe^iomega_ t right^* &qquad timesleftsink_ xe^iomega_ t+sink_ xe^iomega_ t right fracA^leftsink_ xe^-iomega_ t+sink_ xe^-iomega_ t right &qquad times leftsink_ xe^iomega_ t+sink_ xe^iomega_ t right fracA^ sin^k_ x+sink_ xsink_ x e^iomega_-omega_t &qquad + sink_ xsink_ x e^-iomega_-omega_t + sin^k_ x fracA^sin^k_ x+sin^k_ x &qquad + sink_ xsink_ xe^iomega_-omega_t+e^-iomega_-omega_t fracA^sin^k_ x+sin^k_ x &qquad + sink_ xsink_ xcosomega_-omega_t with k_n_pi/L. vspacemm The probability density obviously varies as a function of time which is why a superposition of eigenstates is also referred to as as non-stationary state. The oscillation has a frequency that corresponds to the energy difference between the two levels. vspacemm For an animated graph of the oscillation for n_ and n_ see link for this exercise.

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Branches	quantum physics
Tags	energy level, probability density
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Difficulty	(2, default)
Points	0 (default)
Language	(English)
Type	Calculative / Quantity
Creator	by
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Link	Probability Density 1 – 2