Question

3. At \(t = 0\), a quantum particle bounded in an infinite square well of width \(a\) is prepared in a state described by the wave function \(\Psi(x, 0) = \begin{cases} Ax(a + x)(a - x) & ; \ 0 < x < a, \\ 0 & ; \text{elsewhere}, \end{cases}\) where \(A\) is a positive real constant. The wavefunction \(\Psi(x, 0)\) can be expressed as a linear combination \(\Psi(x, 0) = \sum_{n=1}^{\infty} c_n \psi_n(x),\) where \(\psi_n(x)\) are the stationary state wave functions of the infinite square well with walls at \(x = 0\) and at \(x = a\). (a) (5 pts) Obtain the explicit expression for \(c_n\). (b) (2 pts) What is the wave function \(\Psi(x, t)\) at a later time \(t\)? Express the wave function in terms of the explicit forms of \(\psi_n(x)\) and energies \(E_n\) of a particle in this potential well. (c) (3 pts) What is the expectation value of energy \(\langle E \rangle\)? Does it depend on time \(t\) or not? Explain your answer.

          3. At \(t = 0\), a quantum particle bounded in an infinite square well of width \(a\) is prepared in
a state described by the wave function
\(\Psi(x, 0) = \begin{cases} Ax(a + x)(a - x) & ; \ 0 < x < a, \\ 0 & ; \text{elsewhere}, \end{cases}\)
where \(A\) is a positive real constant. The wavefunction \(\Psi(x, 0)\) can be expressed as a linear
combination
\(\Psi(x, 0) = \sum_{n=1}^{\infty} c_n \psi_n(x),\)
where \(\psi_n(x)\) are the stationary state wave functions of the infinite square well with walls
at \(x = 0\) and at \(x = a\).
(a) (5 pts) Obtain the explicit expression for \(c_n\).
(b) (2 pts) What is the wave function \(\Psi(x, t)\) at a later time \(t\)? Express the wave function
in terms of the explicit forms of \(\psi_n(x)\) and energies \(E_n\) of a particle in this potential well.
(c) (3 pts) What is the expectation value of energy \(\langle E \rangle\)? Does it depend on time \(t\) or not?
Explain your answer.

3. At t = 0, a quantum particle bounded in an infinite square well of width a is prepared in
a state described by the wave function
Ψ(x, 0) = Ax(a + x)(a - x) ; 0 < x < a,
0 ; elsewhere,
where A is a positive real constant. The wavefunction Ψ(x, 0) can be expressed as a linear
combination
Ψ(x, 0) = ∑n=1^∞ cn (x),
where (x) are the stationary state wave functions of the infinite square well with walls
at x = 0 and at x = a.
(a) (5 pts) Obtain the explicit expression for cn.
(b) (2 pts) What is the wave function Ψ(x, t) at a later time t? Express the wave function
in terms of the explicit forms of (x) and energies En of a particle in this potential well.
(c) (3 pts) What is the expectation value of energy ⟨ E ⟩? Does it depend on time t or not?
Explain your answer.

Added by Roberto J.

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