PHYS 507 Homework III 1. (a) Check that the matrix σx is unitary. (b) In the 1st homework, you have shown that there is a hermitian operator B such that σx = eiB . Find a possible solution for B. (Hint: Spectral decomposition.) (c) Coupled with the fact that e2πi = 1, there are many possible solutions for B. Find 0 another matrix B 0 such that B 0 6= B and eiB = eiB = σx . ˜ 0 ) = hp0 |ψi. 2. Remember that we define the momentum space wavefunction by ψ(p ˜ 0 ). (a) Find the following in terms of ψ(p hp0 |ˆ p|ψi =? 0 hp |ˆ x|ψi =? These expressions will tell you how the momentum and position operators look like for the case of momentum-space wavefunction. (b) Use the expressions of the momentum and position operators that you have found in part (a) to show that [ˆ x, pˆ] = i~ . (c) Find hp0 |Tˆ(a)|ψi ˜ 0 ). and express it in terms of ψ(p (c0 ) Let |φi = Tˆ(a) |ψi be the state obtained by translating the state |ψi by a distance ˜ 0 ) and ψ(p ˜ 0 )? Argue that the both states have a. What is the relation between φ(p identical momentum distribution. (In other words, momentum distribution does not change under translation.) 3. Consider the state |ψi given by 0 ψ(x0 ) = N e−α|x | where N and α are real positive constants. (a) Find N so that |ψi is normalized. Page 2 of 2 PHYS 507 - HW 3 (b) Compute hxi, hpi, ∆x and ∆p by using the position space wavefunction. Check the uncertainty relation. Note: In computing ∆p, there are two alternative procedures for finding hp2 i. Z +∞ 2 2 ψ ∗ (x0 )ψ 00 (x0 )dx0 , p = −~ −∞ Z +∞ 2 2 |ψ 0 (x0 )| dx0 . p = ~2 −∞ Do these two expressions give the same value? (Normally they should produce the same value. If they don’t, then that means that something that you are doing is wrong. Perhaps, there is something important that you are missing.) ˜ 0 ) = hp0 |ψi. (c) Now, obtain the momentum space wavefunction ψ(p (d) Plot the probability distribution function for momentum. Show that hp4 i = ∞. (Could you see this from the position-space wavefunction alone?) Z ∞ n! xn e−λx dx = n+1 . Hint: λ 0 4. 0 (a) Show that if position-space wavefunction ψ(x0 ) is real-valued (i.e., ψ ∗ (x0) = ψ(x 2 ), ˜ 0 then positive and negative momentum values are equally probable, i.e., ψ(p ) = 2 ˜ ψ(−p0 ) . (b) Show that, if the position-space wavefunction is real, then average momentum is zero, hpi = 0. 0 (c) Suppose that the position-space wavefunction is of the form ψ(x0 ) = f (x0 )eikx where k is a real constant and f (x0 ) is a real-valued function. Show 2 that the mo2 ˜ ˜ mentum distribution is symmetric around ~k, (i.e., ψ(~k + q) = ψ(~k − q) ) and the average momentum is hpi = ~k. (d) Inversion is a symmetry operation that changes x to −x and p to −p. A function f is even under inversion if f (−x0 ) = f (x0 ); and it is odd under inversion if f (−x0 ) = −f (x0 ). Show that, if the position-space wavefunction has an even or odd inversion symmetry, i.e., if psi(−x0 ) = ±ψ(x0 ), then the corresponding momentum-space wavefunction has also the same symmetry. 5. Consider the Gaussian wavefunction (x0 − a)2 1 0 0 √ exp − + ikx ψ(x ) = 4σ 2 (2π)1/4 σ Find the momentum-space wavefunction and show that the momentum is also distributed like a Gaussian.