Unformatted text preview: PHY4221: Homework 10 Due Nov. 20, 2008 1. A 1D particle of mass m is conﬁned between two inﬁnite potential walls located at
x = ±a, and in the region |x| < a, the potential is V (x) = g sin(πx/a). By treating V (x) as
a perturbation, ﬁnd the correction of energy of the ground state up to the second order in
g . Work out also the ﬁrst order correction to the energy eigenfunction of the ground state.
2. The Hamiltonian of an oscillator is perturbed by an xn (n is a positive integer) nonlinear
term is given by,
H = ω (a† a + 1/2) + g (a + a† )n .
By perturbation theory, calculate the ground state energy eigenvalues up to ﬁrst order in
g . If the ﬁrst order correction of the ground state vector is denoted by |φ , calculate the
inner product n|φ . Finally, compare your perturbation results of energy eigenvalues with
the exact values for the n = 1 case.
3. Suppose that the Hamiltonian of a spin-j particle given by
H = αJ 2 + βJz + εJx where α and β are positive constants, and ε is a small parameter. By ﬁrst order perturbation
theory, determine the ground state energy and ground state vector. 1 ...
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