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Chapter 2: Problem 18

(a) Show that the variables in the Schrödinger equation for a cubic box may be separated and the overall wavefunctions expressed as \(X(x) Y(y) Z(z)\) (b) Deduce the energy levels and wavefunctions. (c) Show that the function .are orthonormal. (d) What is the degeneracy of the level with \(E=14\left(h^{2} / 8 m L^{2}\right) ?\)

Short Answer

Expert verified
The variables in the given Schrödinger equation can indeed be separated. Achieving separation yielded the quantized energy levels and the corresponding wave functions, which can be proven to be orthonormal. The degeneracy level for energy \(E=14\left(h^{2} / 8 m L^{2}\right)\) can be determined by finding the number of different sets of quantum numbers that results in this energy value.

Step by step solution

01

Separation of Variables

To do separation of variables, we start with the Schrödinger equation, which can be written as \(\nabla ^2 \psi + \frac {2m}{\hbar ^2 }(E - V)\psi = 0\). Solving this equation involves separating it into 3 equations, each including only one of the Cartesian coordinates \(x\), \(y\), \(z\). This leads to \(\psi (x,y,z) = X(x) Y(y) Z(z)\).
02

Energy Levels and Wavefunctions

To solve for the energy levels and wavefunctions, we solve each of the above separated equations for Cartesian coordinates, \(X\), \(Y\), \(Z\). The solutions give us quantized energy levels as well as the corresponding wave functions which are product of the solutions of the separated equations.
03

Proving Orthonormality

Orthonormal probability amplitude nature of the wavefunctions can be demonstrated by applying the condition of orthonormality in quantum mechanics; mathematically it involves performing the integral over the product of two different wave functions, including their complex conjugates. If the functions are normalized, this will yield 0. This step confirms the validity of the solutions obtained.
04

Degeneracy Level

The degeneracy of a level with \(E=14\left(\hbar^{2} / 8 m L^{2}\right)\) can be found by identifying all sets of quantum numbers (n_x, n_y, n_z) that lead to this energy level. Calculate and pair up similar energy levels to find the degeneracy.

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Most popular questions from this chapter

An electron is confined to a one-dimensional box of length \(L .\) What should be the length of the box in order for its zero-point energy to be equal to its rest mass energy \(\left(m_{\mathrm{e}} c^{2}\right) ?\) Express the result in terms of the Compton wavelength, \(\lambda_{\mathrm{C}}=h / m_{\mathrm{e}} c\)

A very simple model of a polyene is the free electron molecular orbital (FEMO) model. Regard a chain of \(N\) conjugated carbon atoms, bond length \(R_{\mathrm{CC}},\) as forming a box of length \(L=(N-1) R_{\mathrm{CC}} .\) Find an expression for the allowed energies. Suppose that the electrons enter the states in pairs so that the lowest \(\frac{1}{2} N\) states are occupied. Estimate the wavelength of the lowest energy transition, taking \(R_{\mathrm{CC}}=140 \mathrm{pm}\) and \(N=22 .\) Repeat the calculation of the wavelength if the length of the chain is taken to be \((N+1) R_{\mathrm{CC}}(\) an assumption that allows for electrons to spill over the ends slightly.

The root mean square deviation of the particle from its mean position is \(\Delta x=\left\\{\left\langle x^{2}\right\rangle-\langle x\rangle^{2}\right\\}^{1 / 2} .\) Evaluate this quantity for a particle in a well and show that it approaches its classical value as \(n \rightarrow \infty\). Hint. Evaluate \(\left\langle x^{2}\right\rangle=\int_{0}^{L} x^{2} \psi^{2}(x) \mathrm{d} x\) In the classical case the distribution is uniform across the box, and so in effect \(\psi(x)=1 / L^{1 / 2}.\)

For a particle in a box, the mean value and mean square value of the linear momentum are given by \(\int_{0}^{L} \psi^{*} p \psi \mathrm{d} x\) and \(\int_{0}^{L} \psi^{*} p^{2} \psi \mathrm{d} x,\) respectively. Evaluate these quantities. Form the root mean square deviation \(\Delta p=\left\\{\left\langle p^{2}\right\rangle-\langle p\rangle^{2}\right\\}^{1 / 2}\) and investigate the consistency of the outcome with the uncertainty principle. Hint. Use \(p=(\hbar / \mathrm{i}) \mathrm{d} / \mathrm{d} x .\) For \(\left\langle p^{2}\right\rangle\) notice that \(E=p^{2} / 2 m\) and we already know \(E\) for each \(n\). For the last part, form \(\Delta x \Delta p\) and show that \(\Delta x \Delta p \geq \frac{1}{2} \hbar,\) the precise form of the principle, for all \(n\) evaluate \(\Delta x \Delta p\) for \(n=1.\)

Demonstrate that accidental degeneracies can exist in a rectangular infinite square-well potential provided that the lengths of the sides are in a rational proportion, that is when \(L_{1}=\lambda L_{2},\) with \(\lambda\) an integer. What is the degeneracy?
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