124994 Midterm Exam Fall 2009

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<ul><li><p>8/3/2019 124994 Midterm Exam Fall 2009</p><p> 1/3</p><p>PHYS M1410: Quantum Mechanics I.</p><p>Midterm Examination</p><p>PHYS M1410: Quantum Mechanics (I)November 20, 2009 Friday</p><p>Problem 1. (16 points) (a) Study the hermiticity of the operators:</p><p>, = ,d</p><p> X D P i Ddx</p><p> . What are the complex conjugate of these operators? (b) Show</p><p>that [ , ] X P i . (c) Show that the translation operator /( ) iPaT a e is unitary and is defined</p><p>to be such that ( ) ( ) ( )T a x x a . (d) Consider the following two matrices:</p><p>1 0 1 1 1 2</p><p>= 0 0 0 , 1 0 1</p><p>1 0 1 2 1 1</p><p>. Can they be simultaneously diagonized? If so, give the</p><p>reason why, then find the eigenvectors common to both, and verify that under a unitary</p><p>transformation to this basis, both matrices are diagonized.</p><p>Problem 2. (12 points) (a) Show that the time derivative of the expectation value of any</p><p>observable = | | is given by ,d i</p><p>Hdt t</p><p>, where H is the</p><p>Hamiltonian of the quantum system. (b) Using the result of (a), show that ifH is not explicitly</p><p>time-dependent, the total energy of the system is conserved. (c) If2</p><p>( )2P H V X </p><p>m , show that</p><p>d X</p><p>dtand</p><p>d P</p><p>dthave respective form reminiscent of Hamilton equation of classical</p><p>mechanics.</p><p>Problem 3. (12 points) In a double-slit experiment with a source of monoenergetic electrons,</p><p>detectors are placed along a vertical screen parallel to the y-axis to monitor the diffraction</p><p>pattern of the electrons emitted from the two slits. (a) If the intensity of the electron beam is so</p><p>low such that one is doing the experiment with only one electron at a time, can one predict the</p><p>vertical location of a given single electron on the screen? Why? (b) Can we predict the</p><p>interference pattern if we wait until many electrons have arrived? Why? (c) If a light source is</p><p>used to determine which of the slits the electron went through, can the interference pattern be</p><p>observed on the screen after we wait until many electrons have arrived? Why? [Please give</p><p>your reasons why in (a), (b) and (c) in terms of interpretations by quantum mechanics]..</p><p>Problem 4. (20 points) Consider the following operators on a Hilbert space 3V ( )C </p></li><li><p>8/3/2019 124994 Midterm Exam Fall 2009</p><p> 2/3</p><p>PHYS M1410: Quantum Mechanics I.</p><p>0 1 0 0 0 1 0 01 1 1</p><p>1 0 1 , 0 , 0 0 02 2 2</p><p>0 1 0 0 0 0 0 1</p><p> x y z</p><p>i</p><p> L L i i L</p><p>i</p><p>(a) What are the possible</p><p>values one can obtain if xL is measured? Why? (b) Take the state in which 1xL . In this</p><p>state what are2</p><p>, , andy y L L L , where2 2 2 2</p><p> x y z L L L L . (c) Find the normalized</p><p>eigenstates and eigenvalues of yL in the zL basis. (d) If the particle is in the state with</p><p>1zL , and yL is measured, what are the possible outcomes and their possibilities? (e)</p><p>Consider the state</p><p>2 / 3</p><p>= 1/ 3</p><p>2 / 3</p><p>in thez</p><p>L basis. Ifx</p><p>L is measured in this state and a result</p><p>1 is obtained, what is the state after the measurement? How probable was the result? Ifz</p><p>L </p><p>is measured, what are the possible outcomes and their probabilities?</p><p>Problem 5. (24 points) Consider a free particle with2/ 2 H P m . (a) Find its eigenvalues</p><p>and eigenfunctions in the X basis. (b) Find the propagator ( , ; ', ') ( ) 'U x t x t x U t x (c) </p><p>consider as an initial wave function the wave packet2 2</p><p>0 / / 2 2 1/ 4( ,0) /( )ip x x x e e . Calculate</p><p>(0) , (0) , (0), (0) X P X P . (d) Find ( , )x t and probability density at time t. (e) Find .</p><p>( ) , ( ) , ( ), ( )X t P t X t P t . (f) For a macroscopic particle with 132g and 4 10 cmm ,</p><p>estimate how long is it for ( ) 1mmX t ?</p><p>Problem 6. (16 points) (a) Consider the one-dimensional potential barrier as shown in the</p><p>figure. Write down the wave function that is</p><p>in physical Hilbert space in each region of</p><p>the potential for V2</p></li><li><p>8/3/2019 124994 Midterm Exam Fall 2009</p><p> 3/3</p><p>PHYS M1410: Quantum Mechanics I.</p><p>state, the probability current density is a constant in coordinate (independent of x). (d)</p><p>Calculate the probability current density in Region I and III, as well as using the results of (c),</p><p>show that R+T=1, where R is the reflection coefficient and Tis the transmission coefficient.</p><p>Problem 7. (20 points) Consider a Harmonic oscillator with</p><p>2 2 2</p><p>+2 2</p><p>P m X</p><p>H</p><p> . (a) Show that</p><p>( 1/ 2) H a a , where 1 1</p><p>, and2 2 2 2</p><p>m ma X i P a X i P</p><p>m m</p><p> . Show</p><p>also that [ , ] 1a a . (b) Given that 1 , and 1 1a n n n a n n n , show that</p><p>( 1/ 2) H n n n . (c) Find the normalized ground state wavefunction by projecting</p><p>0 0a on the X basis. (d) Suppose at t=0, a particle starts out in</p><p>1 1 1(0) 0 1 2</p><p>2 2 2 . Find (0)X and the average energy of the system at t=0. (e)</p><p>Find ( ) , ( )t X t , and the average energy at time t.</p><p>Useful Gaussian Integrals:</p><p>2</p><p>2</p><p>2 2</p><p>1/ 2</p><p>1/ 2</p><p>2</p><p>1/ 2</p><p>/ 4</p><p>for 0,</p><p>1,</p><p>2</p><p>.</p><p>x</p><p>x</p><p>x x</p><p>e dx</p><p> x e dx</p><p>e dx e</p></li></ul>

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