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HW5 - a with eigenvalue z(Hint you may find it useful to...

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Quantum Mechanics I, Physics 3143: Assignment 5, Fall 2010 1 . (a) Show that the wave function, Ψ z ( x, t ) = N e - iωt/ 2 X n =0 ( ze - iωt ) n n ! u n ( x ) , with z a complex number, satisfies the time-dependent Schrodinger equation i ~ Ψ ∂t = ˆ H Ψ , for the harmonic oscillator whose Hamiltonian ˆ H = ~ ω ( ˆ a ˆ a + 1 / 2 ) has eigenfunctions u n , n = 0 , 1 , 2 , ... (b) Find the normalization constant N as a function of z . (c) In an energy measurement, under what conditions are the probabilities to measure the ground and first excited state energies equal ? (d) Find the expectation values of the position and momentum operators at time t . NOTE : ˆ au n = nu n - 1 , ˆ a u n = n + 1 u n +1 . ˆ x = r ~ 2 ( ˆ a + ˆ a ) , ˆ p = 1 i r m ~ ω 2 ( ˆ a - ˆ a ) The exponential function satisfies e x = X n =0 x n n ! . 2. (a) Show that Ψ z ( x, 0) is an eigenstate of the operator ˆ
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Unformatted text preview: a with eigenvalue z . (Hint: you may find it useful to write out the first few terms in the sum explicitly. ) (b) Recall from class that the operators ˆ a and ˆ a † are hermitian adjoints, i.e., h φ | ˆ aψ i = h ˆ a † φ | ψ i . Hence show that (one line !) h Ψ z (0) | ˆ a † ˆ a Ψ z (0) i = | z | 2 . (c) Find (two lines) h Ψ z (0) | ˆ a † 2 ˆ a 2 Ψ z (0) i . NOTE : The “overlap” notation is h φ ( t ) | ˆ Aψ ( t ) i := Z ∞-∞ φ * ( x,t ) Aψ ( x,t ) dx. h ˆ Bφ ( t ) | ψ ( t ) i := Z ∞-∞ ( Bφ ( x,t )) * ψ ( x,t ) dx. You don’t need to use the integral forms in this problem. 1...
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