- A one dimensional quantum harmonic oscillator is initially in the state, nx, = A (—nk/r4)11 4 ( .17u x +1)2 e-nt (a) Derive the mean energy. (E) of the osvillator. (b) Write down the time evolved wave function, tli(x, t). (c) Now derive the mean momentum of the oscillator, (p).
(6 Marks) - Consider a free particle anywhere on the x-axis given by the wave function, 111(x) - a o E C and 8> 0. (a) Compute and sketch the wave function in the momentum space, ir(k). (b) Does the uncertainty product, Az Ak depend on the parameters, a and 0? Prove this. (6 Marks)
- Consider the delta function barrier, V(x) = a 6(x), a > 0 (a) Do bound states exist? Explain your answer. (b) Derive the reflection coefficient R and sketch it as a function of particle energy. (4 Marks) 4. Let A be a Hermitian operator with eigenkets (1a1)} corresponding to distinct eigenvalues (a) Prove that
is a null operator. (b) Explain the physical significance of
H(A -
rr (A - ai) isi (a • - ai)
Hint: It gets easy if you take A = 5: of a spin 1/2 system.
(4 Marks)
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