Questions 45 - 48#

Q45 Particle in sloping box#

Repeat the particle in box example calculation and with a perturbing potential of the form $$V = b(x-L/2)^2$$ making it zero in the centre of the box. The constant is $$b = 1/4$$. Use python, based on the code in the example or otherwise, to calculate some of the energy corrections $$E(1),\;E(2)$$ etc.

Q46 Diatomic molecule in electric filed#

A heteronuclear diatomic molecule, which can be adequately described as a harmonic oscillator, is placed in an electric field aligned with the molecule’s long axis and so experiences an additional and linear potential of magnitude $$ax$$. Calculate the change in the energy levels and the resulting spectrum. The harmonic oscillator has vibrational frequency $$\omega$$ and reduced mass $$\mu$$ and orthonormal wavefunctions,

$\displaystyle \psi(x,n) = \frac{1}{\sqrt{2^n n!}}\left(\frac{\alpha}{\pi}\right)^{1/4}H_n(x\sqrt{\alpha}) e^{-ax^2/2}$

where $$\displaystyle \alpha = \sqrt{k\mu/\hbar}$$ and $$\displaystyle H_n(x\sqrt{\alpha})$$ is a Hermite polynomial. You should look these up and either use a recursion formula to calculate values or use the formulae directly is you use only a few.

Strategy: Use the perturbation method to calculate the change in energy. In each case use the harmonic oscillator wavefunctions. The Hamiltonian is $$H = H^0 + ax$$ where $$H^0$$ solves the normal harmonic oscillator with energy $$\displaystyle E_n = \hbar \omega (n + 1/2)$$.

Q47 Perturbed harmonic oscillator#

Suppose that a harmonic potential is modified by a perturbing cubic term of magnitude $$bx^3$$, the oscillator now becomes anharmonic. Calculate the energy levels and spectrum.

Q48 Particle on a ring with potential#

The particle on a ring can approximate the energy levels of a cyclic polyene. The potential energy is zero and the Schroedinger equation

$\displaystyle -\frac{\hbar^2}{2\mu}\frac{d^2\psi}{d\varphi^2}=E\psi$

where the angle $$\varphi$$ has values from $$-\pi \cdots \pi$$ radians. The wavefunction is

$\displaystyle \psi_n= e^{in\varphi}/\sqrt{2\pi}$

and the quantum numbers are $$n = 0, \pm 1, \pm 2, \cdots$$

(a) Calculate the unperturbed energies $$E_n$$.

(b) Calculate the perturbed energy of the lowest level ($$n = 0$$) to second order, when the potential has the value $$V$$ from $$-a\pi \cdots a\pi$$ where $$a$$ is a fraction $$\lt 1$$. If we were to suppose that our ring was pyridine then the nitrogen would have a different potential to that of the carbons. Call this value $$V$$, and then $$a$$ could be 1/6. Find the energy if $$V = 0.1E_1$$. The figure shows a particle on a ring with a small region of perturbation.

Figure 14. Particle on a ring with a small region of perturbation