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Representations of the Infinite Square WellConsider three particles of mass \(m\) which are each in an infinite square well potential at \(0<x<L\).
The energy eigenstates of the infinite square well are:
\[ E_n(x) = \sqrt{\frac{2}{L}}\sin{\left(\frac{n \pi x}{L}\right)}\]
with energies \(E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}\)
The particles are initially in the states, respectively: \begin{eqnarray*} |\psi_a(0)\rangle &=& A \Big[ 2i \left|{E_4}\right\rangle - 3\left|{E_{10}}\right\rangle \Big]\\[6pt] \psi_b(x,0) &=& B \left[ i \sqrt{\frac{8}{L}}\sin{\left(\frac{4\pi x}{L}\right)} - \sqrt{\frac{18}{L}}\sin{\left(\frac{10\pi x}{L}\right)} \right]\\[6pt] \psi_c(x,0) &=& C x(x-L) \end{eqnarray*}
For each particle:
- Determine the normalization constant.
- At \(t=0\) what is the probability of measuring the energy of the particle to be \(\frac{8\pi^2\hbar^2}{mL^2}\)?
- Find state of the particle at a later time \(t\).
- What is the probability of measuring the energy of the particle to be the same value \(\frac{8\pi^2\hbar^2}{mL^2}\) at a later time \(t\)?
- What is the probability of finding the particle to be in the first half of the well?
assignment Homework
Consider the following wave functions (over all space - not the infinite square well!):
\(\psi_a(x) = A e^{-x^2/3}\)
\(\psi_b(x) = B \frac{1}{x^2+2} \)
\(\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)\) (“sech” is the hyperbolic secant function.)
In each case:
assignment Homework
assignment Homework
assignment Homework
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Find an expression for the entropy \(\sigma\). The temperature is \(kT\).
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