Download Quantum Mechanics HW2: Wavefunctions, Time & Angular Operators, Uncertainty - Prof. Leo Ra and more Assignments Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, Fall 2007 PHYS 5250: Quantum Mechanics - I Homework Set 2 Issued September 10, 2007 Due September 24, 2007 Reading Assignment: Shankar, Ch. 1, 4, 5; Sakurai, Ch. 1, 2.1-2.4; Schiff Ch. 2, 3 1. A wavefunction is generically a complex function and therefore can be written as ψ(r, t) = |ψ|eiφ, where its magnitude and phase are real functions. Using Schrodinger’s equation, derive two equations satisfied by |ψ| and S ≡ φh̄, showing that one is a continuity equation, and the other reduces to Hamilton-Jacobi equation in h̄ → 0 limit. 2. Consider the time evolution operator Û(t), defined by |ψ(t)〉 = Û(t)|ψ(0)〉. (a) Using the formal operator expression for Ût show that it satisfies the Schrodinger’s equation. (b) Using the explicit coordinate representation of U0(x, x ′; t) for a free particle (de- rived in class), show that it satisfies the free Schrodinger’s equation. 3. (Shankar 4.2.1) Consider the following explicit expressions for components of the an- gular momentum operator L̂ in the (so called) L = 1 representation: Lx = 1 21/2 0 1 01 0 1 0 1 0 , Ly = 1 21/2 0 −i 0i 0 −i 0 i 0 , Lz = 1 0 00 0 0 0 0 −1 . (1) (a) What are the possible values one can obtain if Lz is measured? (b) Take the state in which Lz = 1. In this state, compute 〈Lx〉, 〈L2x〉, and variance ∆Lx. (c) Find the normalized eigenstates and the eigenvalues of Lx in the Lz basis. (d) If the particle is in the state Lz = −1, and Lx is measured, what are the possible outcomes and their probabilities? (e) Consider the state |ψ〉 = 1 2 11 21/2 (2) in the Lz basis. If L 2 z is measured in this state and a result +1 is obtained, what is the state after the measurement? How probable was this result? If Lz is measured immediately afterwards, what are the outcomes and respective probabilities? (f) A particle is in a state for which the probabilities are P (Lz = 1) = 1/4, P (Lz = 0) = 1/2, and P (Lz = −1) = 1/4. Give an argument that the most general, normalized state with this property is |ψ〉 = e iδ1 2 |Lz = 1〉+ eiδ2 21/2 |Lz = 0〉+ eiδ3 2 |Lz = −1〉. (3) Compute P (Lx = 0) in this state and thereby show that in contrast to an overall phase factor, relative phases δi are indeed physically observable. 4. Using the well-known expression for the orbital angular momentum, L = r×p derive: (a) the Poisson bracket relation satisfied by its components, and (b) the commutation relation satisfied by its components, in the classical and quantum case, respectively, in the latter case understood as oper- ators. Hint: Take advantage of the canonical Poisson bracket and commutation relations between components of r and p. 5. Estimate the spectrum and the extent (along z) of the corresponding eigenstates of a bouncing ball using: (a) Bohr-Sommerfeld quantization, (b) Minimization of the energy, together with the uncertainty principle p ≈ nh̄/z Hint: Ignore any dissipation or ball’s elastic energy. 6. Show that 〈ψ|p̂|ψ〉 = 0 for any state characterized by a real (as opposed to complex) wavefunction ψ. 7. Consider a free electron whose position at time t = 0 was measured to be exactly x = x0. (a) (Up to a proportionality constant) what is ψ(x, 0+) right after (t = 0+) this measurement? Sketch the amplitude of this state.
