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"# Questions 45 - 48"
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"## Q45 Particle in sloping box\n",
"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.\n",
"\n",
"## Q46 Diatomic molecule in electric filed\n",
"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,\n",
"\n",
"$$\\displaystyle \\psi(x,n) = \\frac{1}{\\sqrt{2^n n!}}\\left(\\frac{\\alpha}{\\pi}\\right)^{1/4}H_n(x\\sqrt{\\alpha}) e^{-ax^2/2}$$\n",
"\n",
"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.\n",
"\n",
"**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 $\\displaystyle E_n = \\hbar \\omega (n + 1/2)$.\n",
"\n",
"## Q47 Perturbed harmonic oscillator\n",
"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.\n",
"\n",
"## Q48 Particle on a ring with potential\n",
"The particle on a ring can approximate the energy levels of a cyclic polyene. The potential energy is zero and the Schroedinger equation \n",
"\n",
"$$\\displaystyle -\\frac{\\hbar^2}{2\\mu}\\frac{d^2\\psi}{d\\varphi^2}=E\\psi$$\n",
"\n",
"where the angle $\\varphi$ has values from $-\\pi \\cdots \\pi$ radians. The wavefunction is \n",
"\n",
"$$\\displaystyle \\psi_n= e^{in\\varphi}/\\sqrt{2\\pi}$$\n",
"\n",
"and the quantum numbers are $n = 0, \\pm 1, \\pm 2, \\cdots$\n",
"\n",
"(a) Calculate the unperturbed energies $E_n$.\n",
"\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.\n",
"\n",
"![Drawing](series-fig14.png)\n",
"\n",
"Figure 14. Particle on a ring with a small region of perturbation"
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