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Hello students, in the question first we need to find the energy eigenvalues and the eigenstates.
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The energy eigenfunction for an infinite potential well is given by psi n x equals to under root 2 upon a sin n pi x upon a where n is the integer, a is the width of the well.
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The energy eigenvalues are given by e n equals to n square into pi square into h square upon 2 m a square where h is planck's constant.
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Now we are given the initial wave function that psi x comma t equals to 0 is equals to 5 a 1 plus cos 7 x sin x.
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We can rewrite this wave function using the hint provided as psi x comma t equals to 0 equals to 5 a 1 plus cos 7 x sin x equals to 5 a 2 sin x cos x this equals to 10 a sin x cos x.
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Now comparing this with the energy eigenfunction we see that this wave function is a superposition of first and second energy eigenfunctions that is psi x comma t equals to 0 equals to a psi 1 x plus b psi 2 x where a and b are coefficient.
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Since the wave function is normalized we can write a and b as a equals to root 2 upon a and b equals to 5 root 2 upon a.
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So the energy eigenvalues of e 1 equals to pi square h square upon 2 m a square e 2 equals to 4 pi square h square upon 2 m a square.
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Now next to find the eigenfunction the wave function at a later time t we use the time determinant schrodinger equation that is psi x comma t equals to a psi 1 x e to the power minus i e 1 t upon h...