Quantum Theory Sheet 1 - MT21 (Questions on sections 0-3 of the lecture notes) 1. The potential energy for an electron in a hydrogen atom is V(r) = e2 4TTEOT where -e is the charge of the electron, r is its distance from the nucleus, and co is a constant. In a circular orbit the electron has angular momentum L = mur, where m is the electron mass and v is its speed. In 1913 Bohr proposed that L is quantized, satisfying L = nh where n is a positive integer. (a) Given that Newton's second law for circular orbits is 2 m = = V' ( r ) , show that Bohr's quantization implies r = n2a, where a me2 4TT Eoh2 is called the Bohr radius. (b) Show that the total energy E = 1mv2 + V(r) is given by 2ma2 E = ?2 · 1 nº . [This successfully reproduces the hydrogen atom energy levels (1.3) in the lecture notes, but a full quantum mechanical treatment will only appear at the end of our course.] 2. A particle of mass m moves in the interval [-a, a] where the potential V = Vo is constant. Using the stationary state Schrödinger equation show that the energy levels of the system are En = V0 + 8ma2 n2+2h2 , where n is a positive integer, and find the corresponding normalized wave functions. Show that the wave functions are all either even or odd functions of x. 3. A particle of mass m moving on the x-axis has a (non-normalized) ground state wave function sech2 x with energy -2h2/m. (a) Show that the potential is V(x) = - 3h2 sech2 x. (b) An excited state wave function for the particle is (x) = tanh x sech x. What is the energy of this state? Mathematical Institute, University of Oxford James Sparks: sparks@maths.ox.ac.uk Page 1 of 2
Quantum Theory: Sheet 1- MT21 4. Consider a particle of mass m confined to a box in three dimensions, with potential 0 0, 0<x<a,0<y<b, 0 <<< c, V(x, y, z) = too otherwise , where (x, y, z) are Cartesian coordinates. By separating variables in the stationary state Schrödinger equation, show that the allowed energies of the particle are En1, n2, n3 TT272 1 ,2 + 2 + ?m C2 , where n1, n2, ng are positive integers, and find the corresponding normalized wave func- tions. [You may use the results for the one-dimensional box.] 5. A particle of mass m moves on the x-axis in a potential V(x), where V is an even function (that is V(x) = V(-x)). Let (x) be a normalized wave function satisfying the stationary state Schrödinger equation with energy E. (a) Show that (x) = {(-x) is also a normalized wave function. (b) By considering the wave functions (+ = { +v, or otherwise, deduce that there is either an even or an odd wave function (or both) satisfying the same Schrödinger equation. 6. Suppose that !(x, t)