10.20. A small perturbation, $W = ax^4$, is applied to a harmonic oscillator with force constant $k$ and reduced mass $m$. Compute the first-order correction to the eigenenergies and first nonvanishing correction to the wave functions.
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Step 1: The Hamiltonian for a harmonic oscillator is given by: $H_0 = \frac{p^2}{2m} + \frac{1}{2}kx^2$ The eigenenergies are given by: $E_n^{(0)} = \hbar\omega(n + \frac{1}{2})$, where $\omega = \sqrt{\frac{k}{m}}$ The eigenfunctions are given by: $\psi_n^{(0)}(x) Show more…
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Consider a simple harmonic oscillator formed by a mass m attached to a spring k and sliding on a frictionless horizontal surface. a. Just for practice, write down the Hamiltonian, as well as the energy eigenvalues and eigenstates of this system. b. Suppose now that a small, constant, positive force F is applied now to the mass, perhaps by tilting the surface to the right slightly. Derive the expression for the perturbation Hamiltonian H'. Using perturbation theory, calculate the first and second order corrections to the energy eigenvalues and the first order correction to the ground state eigenfunction. c. Derive the exact values of the energy eigenvalues and eigenstates of the perturbed system from part (b) and compare them with you've found in part (b). Comment on the correctness of the perturbation theory.
Aarya B.
Consider the solution of the 1D oscillator problem. Compute the correction to the energy eigenvalues caused by the following perturbations: - H' = cx - H' = cx³ - H' = cx⁴ where c is a real constant.
Supreeta N.
The WKB approximation (see Challenge Problem 40.64) can be used to calculate the energy levels for a harmonic oscillator. In this approximation, the energy levels are the solutions to the equation $$\int ^b _a \sqrt{2m[E - U(x)]} dx = {nh \over 2} \space n = 1, 2, 3, . . .$$ Here $E$ is the energy, $U(x)$ is the potential-energy function, and $x = a$ and $x = b$ are the classical turning points (the points at which $E$ is equal to the potential energy, so the Newtonian kinetic energy would be zero). (a) Determine the classical turning points for a harmonic oscillator with energy $E$ and force constant $k'$. (b) Carry out the integral in the WKB approximation and show that the energy levels in this approximation are $E_n = \hslash\omega$, where $\omega = \sqrt{k'/m}$ and $n$ = 1, 2, 3, c. ($Hint$: Recall that $\hslash = h/2\pi$. A useful standard integral is $$\int \sqrt{A^2 - x^2} dx = {1\over2} [ x\sqrt{A^2 - x^2} + A^2 arcsin ({x \over \mid A\mid}) ]$$ where arcsin denotes the inverse sine function. Note that the integrand is even, so the integral from $-x$ to $x$ is equal to twice the integral from 0 to $x$.) (c) How do the approximate energy levels found in part (b) compare with the true energy levels given by Eq. (40.46)? Does the WKB approximation give an underestimate or an overestimate of the energy levels?
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