Unformatted text preview: NAME____________________________________________________________ PHYS 360 Quantum Mechanics Spring 2010 Monday, Mar 29 2010 Mid
Term Exam 2 This is a closed
book exam. You should have your own hand
written page of equations and notes. You may also have the hand
written page you used for Midterm 1. Before starting, make sure you have signed your name above. This exam is worth 100 points. Each question has the point total indicated. Write everything on these pages. Partial credit may be awarded for incorrect or incomplete answers if you have shown sufficient understanding. It is very important that your work is legible. In some cases no points will be awarded for a correct answer if you have not shown your work in solving the problem. Solutions will be posted on the course web site within one day. ALL exams must be turned in no later than 2:20 pm, the end of the scheduled class period. There should be ten numbered pages to this exam. 1 Problem One: Central Potential in Two Dimensions In three dimensions we used spherical coordinates and had a differential volume element dv = (dr ) × (r sin θ dθ ) × (rdφ ) = r 2 sin θ drdθ dφ . We will do this problem in two dimensions with polar coordinates, where the differential area element is da = (dr) × (rdφ ) = rdrdφ. Consider a central potential, where V (r ) = V (r, φ ) = V (r ) , that is, the potential only depends on the distance from the origin. The Schrödinger equation can be written as: ⎛ 2 2 ⎞ ∂Ψ (r , t ) i = H Ψ (r , t ) = ⎜ − ∇ + V (r )⎟ Ψ (r , t ) ∂t ⎝ 2m ⎠ =− where the definition of the ∇ 2 operator in polar coordinates has been inserted: 1 ∂ ⎛ ∂ ⎞ 1 ∂2 ∇2 = . ⎜r ⎟ + r ∂r ⎝ ∂r ⎠ r 2 ∂φ 2 Question 1: (5 points) We apply separation of variables to get the time dependence of each eigenstate as Ψ nm (r , t ) = ψ nm (r )φnm (t ) = ψ nm (r )e− iEnm t / . This is allowed because: [ ] A. the potential is real. [ ] B. the wave functions are square integrable. [ ] C. these are determinate states. [ ] D. the potential commutes with the Hamiltonian. [ ] E. the potential is independent of time. 2 ⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂2 ⎤ ⎢ r ∂r ⎜ r ∂r ⎟ + r 2 ∂φ 2 ⎥ Ψ (r , t ) + V (r )Ψ (r , t ) ⎝ ⎠ 2m ⎣ ⎦ 2 Problem One: Central Potential in Two Dimensions Question 2: (10 points) The time
independent Schrödinger equation can now be written as 2 ⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂2 ⎤ − ψ nm (r, φ ) + V (r )ψ nm (r, φ ) = Enmψ nm (r, φ ). ⎜r ⎟ + 2 m ⎢ r ∂r ⎝ ∂r ⎠ r 2 ∂φ 2 ⎥ ⎣ ⎦ Apply separation of variables, ψ nm (r, φ ) ≡ Rnm (r )Φ nm (φ ) , and derive a) the equation for R(r ), and b) the equation for Φ(φ ). 3 Problem One: Central Potential in Two Dimensions Question 3: (10 points) What are the solutions for Φ(φ ) ? That is, find the functions that solve the equation for Φ . Also, use the requirement that the functions must be “single
valued,” which means Φ(φ ) = Φ(φ + 2π ) , to determine the allowed values of the quantum number. 4 Problem One: Central Potential in Two Dimensions Question 4: (5 points) Write down the normalization condition for the wave function ψ nm (r, φ ) ≡ Rnm (r )Φ nm (φ ) . That is, give the definite integral that must equal 1, with the specific limits of integration. 5 Problem Two: One
dimensional Potential Barrier A one
dimensional square potential barrier is given by: Region I: VI ( x ) = 0 x<0 $#
II! I! Region II: VII ( x ) = +V 0 ≤ x !" # We consider a system with E > V . Question 1: (5 points) Write down the time
independent Schrödinger equations for region I and region II. Question 2: (5 points) Write down the general solution for the eigenfunctions ψ I ( x ) in region I. That means, don’t worry yet about boundary conditions, normalizations, etc. 6 Problem Two: One
dimensional Potential Barrier Question 3: (5 points) Write down the general solution for the eigenfunctions ψ II ( x ) in region II. Now we consider a specific situation. Assume a traveling wave of amplitude A exists in region I and is moving in the +x direction, towards the barrier. Also assume the energy eigenvalue is larger than the potential barrier: E > V . Question 4: (10 points) What are the boundary conditions at x=0? Apply these to your functions from the previous questions. 7 Problem Two: One
dimensional Potential Barrier Question 5: (10 points) What is the intensity of the reflected wave, as a fraction of the intensity of the incident wave? 8 Problem Three: Commutator for x and p ˆ ˆ (15 points) For the position operator x and the momentum operator px , derive the ˆˆ commutation relation, [ x, px ] = i. 9 Problem Four: Multiple Choice Question 1: (5 points) Assume you have an ensemble of identically prepared particles, for example, a container of hydrogen atoms each of which is described by the same wave function Ψ (r , t ). The expectation value of the energy is E . This means that [ ] A. the most likely result of measuring a particle’s energy is E . [ ] B. every particle actually has the energy E . [ ] C. each particle is in a determinate state with energy E . [ ] D. (none of the above). Question 2: (5 points) Any operator will commute with itself. Consider the fact that the Hamiltonian operator commutes with itself. One consequence of this is [ ] A. the Hamiltonian is Hermite. [ ] B. energy is conserved. [ ] C. energy is conserved if the potential is independent of time. [ ] D. the Hamiltonian must also commute with the momentum operator. Question 3: (5 points) We generally need to construct “wave packets” to describe free particles (meaning particles in a potential V=0). [ ] A. This is because a single eigenfunction of the Hamiltonian cannot describe a particle localized within any finite region. [ ] B. This is because a single eigenfunction of the Hamiltonian cannot specify a determinate momentum (i.e. a momentum of a specific value). [ ] C.This is necessary to make the phase velocity equal the group velocity. [ ] D. (none of the above). Question 4: (5 points) The wave functions for the hydrogen atom are: ψ nlm
⎛ 2 ⎞ ( n − l − 1) ! − r / na ⎛ 2r ⎞ 2 l +1 m =⎜ ⎟ e ⎜ na ⎟ ⎡ Ln−l −1 ( 2r / na ) ⎤ Yl (θ , φ ) ⎦ ⎝ na ⎠ 2 n[( n + l ) !]3 ⎝ ⎠⎣
3 l [ ] [ ] [ ] [ ] A. These go to zero for r → ∞ and for r → 0 , necessary for normalization. B. These go to zero for r → ∞ and for r → 0 , except for the ground state. C. There are (infinitely) many wave functions that do not go to zero as r → 0 . D. Coulomb repulsion ensures the wave function is zero at the origin. 10 ...
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 Spring '11
 DURBIN,STEPHEN
 mechanics, New Mexico, the00

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