PHY413 Quantum Mechanics B Duration: 2 hours 30 minutes

Transcription

1 BSc/MSci Examination by Course Unit Thursday nd May 4 : - :3 PHY43 Quantum Mechanics B Duration: hours 3 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY AN INVIGILATOR. Instructions: Answer ALL questions from Section A. Answer ONLY TWO questions from Section B. Section A carries 5 marks, each question in section B carries 5 marks. If you answer more questions than specified, only the first answers up to the specified number) will be marked. Cross out any answers that you do not wish to be marked. Only non-programmable calculators are permitted in this examination. answer book the name and type of machine used. Please state on your Complete all rough workings in the answer book and cross through any work that is not to be assessed. Important note: The academic regulations state that possession of unauthorised material at any time when a student is under examination conditions is an assessment offence and can lead to expulsion from QMUL. Please check now to ensure you do not have any notes, mobile phones or unauthorised electronic devices on your person. If you have any, raise your hand and give them to an invigilator immediately. It is also an offence to have any writing of any kind on your person, including on your body. If you are found to have hidden unauthorised material elsewhere, including toilets and cloakrooms it will be treated as being found in your possession. Unauthorised material found on your mobile phone or other electronic device will be considered the same as being in possession of paper notes. A mobile phone that causes a disruption is also an assessment offence. EXAM PAPERS MUST NOT BE REMOVED FROM THE EXAM ROOM. Examiners: Prof A. Brandhuber Dr A. Misquitta c Queen Mary, University of London, 4

2 Page PHY43 4) SECTION A Answer ALL questions in Section A Question A a) Write down the time-dependent Schrödinger Equation TDSE) for a particle of mass m in a 3D potential V x, t). b) State under what conditions one can derive the time-independent Schrödinger Equation TISE) from the TDSE. Write down the TISE and the relation between solutions of the TISE and solutions of the TDSE. Question A Explain the significance of the Operator Postulate. Using the Operator Theorem derive the Hamiltonian Ĥ for the D simple harmonic oscillator and the z-component of the angular momentum operator ˆL z from the corresponding classical expressions. Question A3 a) Write down without proof all non-vanishing commutation relations between the operators x, y, z, ˆp x, ˆp y, and ˆp z. b) Determine the commutators [ x, p x ] and [x+y, ˆp xˆp y ] using the relations from a) or otherwise. Question A4 At time t = a system is in the state Ψx, ) = c ψ x) + c ψ x) where the ψ n are normalised energy eigenstates with energy eigenvalue E n. No derivations needed, brief answers are sufficient. a) What is the relation between c and c if Ψx, ) is normalised? b) What is the probability of an energy measurement returning the value E? If the system is left unperturbed after such a measurement at t =, write down its wavefunction at t >. c) What is the average energy, E = Ĥ, obtained in an ensemble measurement of energy? Question A5 Assume that two hermitian operators  and B do not commute. Write down Heisenberg s generalised uncertainty relation and explain its significance. State another consequence of the fact that the two operators do not commute.

3 PHY43 4) Page 3 Question A6 a) An operator  has eigenstates φ n with eigenvalues λ n. Write this statement as an equation. b) Assuming  is hermitian what properties does this imply for the eigenvalues and eigenfunctions of Â? Express the property of the eigenfunctions as an equation. Question A7 Which of the following operators or matrices are Hermitian? No derivations required, just answer Yes/No. ) ) m ˆp x, ˆp x + iˆx, ˆxˆp x, h e iφ e iφ, Question A8 The orbital angular momentum operators are denoted by L x, L y and L z. a) Write down the expressions for the commutators [ L x, L y ], [ L y, L z ], and [ L z, L x ]. b) Using the relations from a) show that L = L x + L y + L z commutes with L z. Question A9 a) Simultaneous eigenstates of the abstract angular momentum operators Ĵz and J have eigenvalues hm j and h jj + ). What are the allowed values of j and m j and how many different values of m j are there for fixed j? b) State an experimental result that motivates the existence of spin. What physical property does the experiment actually measure? Question A a) Sketch the wave functions ψx) for the ground and first excited states of the infinite square well potential. b) Sketch the corresponding distributions of ψx). c) Write down the parity eigenvalues of these states. Turn over

4 Page 4 PHY43 4) SECTION B Answer TWO questions from Section B Question B a) i) Assuming that the Hamiltonian of the system is hermitian, state the expansion theorem for a particle in a -dimensional potential where the normalised energy eigenstates are ψ n x) with corresponding eigenvalues E n. Write down a formula for the expansion coefficients c n and give an expression for the wave function Ψx, t) with t > if Ψx, ) = n c n ψ n x). ii) Hence, derive an expression for the expectation value of the energy E = the E n and c n if the system is in the state Ψx, t). Ĥ in terms of iii) Consider now a general operator  and show that  can be expressed in the form n,m c ma mn c n if the system is in the state Ψx, ) = n c n ψ n x). Find a formula for the matrix form of the operator A mn. Write down the matrix form of the state Ψx, ). ) E b) Matrix Quantum Mechanics: consider a two-state system with Hamiltonian H = E ) 3 4 and another operator S = i) Write ) down the normalised ) eigenvectors ψ, of H. Show that the normalised vectors φ = 5 and φ = 5 are eigenstates of S. Find the corresponding eigenvalues and express φ, in terms of ψ,. Furthermore, find expressions for ψ, in terms of φ, this will be needed in the questions below). ii) Now a sequence of measurements is performed on the system. In the following ignore the time evolution of the system and only take into account the collapse of the wave function after each measurement: A beam of particles with energy E is prepared. Write down the state of these particles. The beam is sent through a first apparatus that measures S: obtain the possible outcomes, their probabilities and the corresponding states after the measurement. All particles that come out of the first apparatus are sent through a second apparatus that measures the energy of the particles. Find the probability to finding E again. [8 marks]

5 PHY43 4) Page 5 Question B a) The Hamiltonian for a one dimensional harmonic oscillator can be written in the form: Ĥ = hω â â + ) where, [â, â ] =. You also may use without proof the identities â ψ n = n + ψ n+ and âψ n = nψ n. i) State the possible eigenvalues of Ĥ. ii) Prove the commutation relations [Ĥ, â ] = hω â and [Ĥ, â] = hω â. [ mark] [3 marks] iii) Hence show that âψ n and â ψ n have energy eigenvalues E n hω provided that n and E n + hω, respectively, where E n is the eigenvalue of ψ n. iv) Using operator formalism and the fact that ˆx = h mω â + â ) find the expectation values x, x and the uncertainty x when the system is in the eigenstate ψ n. b) Consider a particle with mass m and charge q in a constant magnetic field in the z direction B =,, B) with corresponding vector potential A = By/, Bx/, ). If the particle is confined to move in the x-y plane the Hamiltonian is given by Ĥ = ˆΠx ) + ˆΠ m y ) ) with ˆΠ x = ˆp x qa x and ˆΠ y = ˆp y qa y. i) Calculate the commutator [ˆΠ x, ˆΠ y ]. ii) Consider the operators Ĉ = Ĉx + iĉy, Ĉ = Ĉx iĉy, Ĉ x,y = qb h Π x,y. [3 marks] Find the commutator [Ĉ, Ĉ ] and show that the Hamiltonian has the form Ĥ = hωĉ Ĉ + ). What is the cyclotron frequency Ω and what are the energy eigenvalues Ĥ? [9 marks] Turn over

6 Page 6 PHY43 4) Question B3 a) Hydrogen atom: Consider the eigenstates ψ = Rr)Y ll θ, φ) where l is a non-negative integer. i) First consider the case of the groundstate l = with ψ = Ne r/a : Use the formula r= rn e r/c = n!c n+ to find the normalisation constant N, the expectation values r, r and the uncertainty r in terms of the Bohr radius a. Furthermore, find the value of r for which the radial probability r ψ takes the maximum. [8 marks] ii) Now consider the radial Schrödinger equation for the hydrogen atom to find R for general l [ h d m dr + ] d Rr) + h ll + ) Rr) e Rr) = ERr). r dr mr 4πɛ r Using the ansatz R = Nr l e r/b and inserting it in the radial equation, prove that the values of b and E for which the ansatz is an exact solution are given by b = l+)a and E = E /l+) where a = 4πɛ h /me ) is the Bohr radius and E = me 4 / h 4πɛ ) ) is the groundstate energy. [9 marks] b) Spin of the electron: One important correction to the Hamiltonian of the hydrogen atom Ĥ is the spin-orbit coupling which is of the form Ĥ SO = d n,l S L, where d n,l is a known constant. i) Give a brief, physical explanation for the form of this interaction. Obtain [ĤSO, S z ] and [ĤSO, L z ] and, hence, prove that [ĤSO, Ĵz] = where Ĵz = L z + S z. Show that S and L commute with Ĥ SO. You may use without proof the commutation relations of the S and L operators. Also note that [S i, L k ] = for i, k = x, y, z.) [6 marks] ii) Assuming that the electron is in an eigenstate of the operators J = L + S), Ĵz, L, S with eigenvalues jj + ) h, m j h, ll + ) h, 3 4 h find the shift of the energy eigenvalue given by the expectation value ĤSO. [ marks]

7 PHY43 4) Page 7 Question B4 a) The matrices representing the spin angular momentum operators for a given particle are: S x = h, S y = h The normalised eigenstates of S z are: χ + = i i i i, χ =, S z = h, χ = i) Write down the spin s of the particle and the S z eigenvalues of χ +, χ and χ... [ marks] ii) Calculate the expectation values of S x and S x and, hence, find the uncertainty S x for a particle in the state χ +. iii) Obtain the normalised eigenvector χ + which has eigenvalue + h) of the operator S x and express it in terms of χ +, χ and χ. iv) Assume the system is prepared in the state χ + found in iii) and S z is measured using a Stern Gerlach experiment. What are the possible outcomes of this measurement and the respective probabilities? In a Stern-Gerlach experiment, how many dots would you see on the screen? [3 marks] b) Spin precession: Assume that the spin s particle has electric charge q and mass m and is exposed to a constant magnetic field B =,, B). Ignoring all other interactions and assuming the particle is at rest the Hamiltonian is given by Ĥ = µ B with µ = q m S. i) Obtain the time-dependent Schrödinger equation and find its general solution. [6 marks] ii) Now assume that at t = the particle is in the state χ +. Hence, find the expectation values of all three components of the spin operator S x, S y and S z for t > using your results from b)i) and a)iii). What is the precession frequency? End of Paper