Chapter 2

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?

Confirm that the completeness relation, eqn 1.25, may be expressed in terms of wavefunctions as \\[ \sum_{n} \psi_{n}(r) \psi_{n}^{*}\left(r^{\prime}\right)=\delta\left(r-r^{\prime}\right) \\] where \(\delta\left(r-r^{\prime}\right)\) is the Dirac \(\delta\) -function described in Section 2.1

Locate the nodes of the harmonic oscillator wavefunction for the state with \(v=6 .\) Hint. Use mathematical software.

Evaluate the matrix elements (a) \(\langle v+1|x| v\rangle\) and (b) \(\left\langle v+2\left|x^{2}\right| v\right\rangle\) of a harmonic oscillator by using the relations given at the bottom of Table 2.1 for the Hermite polynomials.

The oscillation of the atoms around their equilibrium positions in the molecule HI can be modelled as a harmonic oscillator of mass \(m \approx m_{\mathrm{H}}\) (the iodine atom is almost stationary \()\) and force constant \(k_{\mathrm{f}}=313.8 \mathrm{N} \mathrm{m}^{-1} .\) Evaluate the separation of the energy levels and predict the wavelength of the light needed to induce a transition between neighbouring levels.

Determine the probability of finding the ground-state harmonic oscillator stretched to a displacement beyond the classical turning point. Hint. Relate the expression for the probability to the error function, erf \(z,\) defined as \\[ \operatorname{erf} z=1-\frac{2}{\pi^{1 / 2}} \int_{z}^{\infty} \mathrm{e}^{-y^{2}} \mathrm{d} y \\] and evaluate it using erf \(1=0.8427 .\) The error function is incorporated into most mathematical software packages.

Calculate the values of (a) \(\langle x\rangle\) (b) \(\left\langle x^{2}\right\rangle\) \(,(\mathrm{c})\left\langle p_{x}\right\rangle,(\mathrm{d})\left\langle p_{x}^{2}\right\rangle\) for a harmonic oscillator in its ground state by evaluation of the appropriate integrals (as in Problems \(2.13-2.15\) ). Examine the value of \(\Delta x \Delta p_{x}\) in the light of the uncertainty principle. Hint. Use the integrals \\[ \begin{array}{l} \int_{-\infty}^{\infty} \mathrm{e}^{-\alpha x^{2}} \mathrm{d} x=\left(\frac{\pi}{\alpha}\right)^{1 / 2} \\ \int_{0}^{\infty} x \mathrm{e}^{-\alpha x^{2}} \mathrm{d} x=\frac{1}{2 \alpha} \\\ \int_{-\infty}^{\infty} x^{2} \mathrm{e}^{-\alpha x^{2}} \mathrm{d} x=\frac{1}{2}\left(\frac{\pi}{\alpha^{3}}\right)^{1 / 2} \end{array} \\]

2.32 Consider a harmonic oscillator of mass \(m\) undergoing harmonic motion in two dimensions \(x\) and \(y .\) The potential energy is given by \(V(x, y)=\frac{1}{2} k_{x} x^{2}+\frac{1}{2} k_{y} y^{2}\). (a) Write down the expression for the hamiltonian operator for such a system. (b) What is the general expression for the allowable energy levels of the two- dimensional harmonic oscillator? (c) What is the energy of the ground state (the lowest energy state)? Hint. The hamiltonian operator can be written as a sum of operators. 2.33 Consider a particle of mass \(1.00 \times 10^{-25} \mathrm{g}\) moving freely in a three-dimensional cubic box of side \(10.00 \mathrm{nm}\) The potential energy is zero inside the box and is infinite at the walls and outside of the box. (a) Evaluate the zero-point energy of the particle. (b) Consider the energy level that has an energy 9 times greater than the zero-point energy. What is the degeneracy of this level? Identify all the sets of quantum numbers that correspond to this energy. (The energy levels of the cubic box were deduced in Problem \(2.18 .\) (c) Compute the wavelength, frequency, and wavenumber of the photon responsible for the transition from the ground state of the particle to the energy level of part (b).

Consider a harmonic oscillator of mass \(m\) undergoing harmonic motion in two dimensions \(x\) and \(y .\) The potential energy is given by \(V(x, y)=\frac{1}{2} k_{x} x^{2}+\frac{1}{2} k_{y} y^{2}\). (a) Write down the expression for the hamiltonian operator for such a system. (b) What is the general expression for the allowable energy levels of the two- dimensional harmonic oscillator? (c) What is the energy of the ground state (the lowest energy state)? Hint. The hamiltonian operator can be written as a sum of operators.