Download 5 Problems on Quantum Mechanics - Assignment 2 | PHYSICS 137B and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 137B, Fall 2007 Problem set 2: perturbation theory and some identical particles review Assigned Friday, 7 September. Due (in 251 LeConte box) 5 pm Friday, 10 September. 1. (from Ohanian) Suppose that the electron in a hydrogen atom is perturbed by a repulsive potential concentrated at the origin. Assume that the potential has the form of a delta function, so that the perturbed Hamiltonian is H = p2 2m − 1 4π0 e2 r +Aδ(r). (1) (a) To first order in the constant A, find the change in the energy of the state with quantum numbers n ≥ 1, l = 0. Hint: ψn00(0) = 2√4π (na0) −3/2. (b) Find the change in the wavefunction to first order in A, using the nondegenerate perturbation theory formula. You may leave the answer in the form of an infinite series, but make sure that all the terms in your series are necessary. 2. (from Ohanian) A particle of mass m is confined to a one-dimensional infinite square potential well that extends from x = 0 to x = L (i.e., V = 0 in the well, V = ∞ outside). Impose appropriate boundary conditions to show that the energy eigenstates for the (nonrelativistic) Hamiltonian are En = n2π2h̄2 2mL2 . (2) If the mass of the particle is small or if the length L is small, the energy eigenstates will be large, and the particle may become relativistic (this happens if the energy is large compared with mc2). The relativistic kinetic energy is H = √ p2c2 +m2c4 −mc2 (3) (a) Without approximating the square root, use first-order perturbation theory to find the new energy eigenvalues that correspond to this relativistic Hamiltonian (relativistic particle in well). (b) The energy eigenvalues obtained by first order perturbation theory are actually the exact eigenvalues for the relativistic Hamiltonian. Explain why this is so. 3. The point of this problem and the next is to understand how to make a Hilbert space for a particle with spin. The same idea is used to make a Hilbert space for multiple particles. We define the tensor product of two complex vector spaces A and B as the complex vector space spanned by “product states” |ψi ⊗ χj〉, alternately written |ψi〉 ⊗ |χj〉, where |ψi〉 ranges over a basis of A and |χj〉 ranges over a basis of B. For the case where ψi is an orbital wavefunction and χj describes spin, this product state is interpreted as describing a particle with orbital state |ψi〉 and spin state |χj〉. An example: the total state of an electron lives in the tensor product of orbital vector space and spin vector space, which for a spin-half particle like the electron has dimension 2. In order to work in the tensor product space, we need to define an inner product. We defined the inner product of two product states 〈ψi ⊗ χj |ψk ⊗ χl〉 to be 〈ψi|ψk〉 × 〈χj |χl〉, where now × is just ordinary multiplication. 1
