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Homework Week 3: Due Friday, October 18 Chemistry 110A 2013 Professor McCurdy 1. [Note that this problem can be done either using Dirac notation or...

I need help with question 2 in which I need to find the normalization constant using Dirac notation.

1 Homework Week 3: Due Friday, October 18 Chemistry 110A — 2013 Professor McCurdy 1. [Note that this problem can be done either using Dirac notation or not. ] The following wave function (in which A and B are numbers – not functions of x or t ) Ψ x , t ( ) = A ψ 1 x ( ) e iE 1 t / h + B ψ 2 x ( ) e iE 2 t / h or Ψ ( t ) = A ψ 1 e iE 1 t / h + B ψ 2 e iE 2 t / h is a linear combination of two solutions of the time dependent Schrödinger equation each having the form ψ n ( x ) e iE n t / h or ψ n e iE n t / h with n=1 and 2, each of which corresponds to a solution of the time independent equation ˆ H ψ n ( x ) = E n ψ n ( x ) or ˆ H ψ n = E n ψ n Problem: Show that " x , t ( ) is a solution of the time-dependent Schrödinger equation. Hint: The algebra is short. All you need is that these functions, " n ( x ) , are eigenfunctions of H . Use that fact each time the combination ˆ H ψ n ( x ) (or ˆ H ψ n ) appears in ˆ H Ψ ( x , t ) = i h ∂Ψ ( x , t ) / t ( or ˆ H Ψ ( t ) = i h t Ψ ( t ) ) 2.[A little easier if you use Dirac notation] Suppose that for the particle-in-box, the system is prepared in the state described by the normalized time-dependent wave function " x , t ( ) : Ψ x , t ( ) = N ψ 1 x ( ) e iE 1 t / h + ψ 2 x ( ) e iE 2 t / h ( ) or Ψ ( t ) = N ψ 1 e iE 1 t / h + ψ 2 e iE 2 t / h ( )
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2 where " n x ( ) = 2 L sin n # x L $ % & ' ( ) and E n = n 2 " 2 ! 2 2 mL 2 are the normalized particle-in-a-box functions and their corresponding energies. (a) Find the normalization constant N, and show that it does not depend on time. Hint: use the fact that the solutions, " n x ( ) , are orthonormal, which makes the algebra short. (b) Evaluate the expectation value of the energy E . Does it depend on time? Should it? Why or why not? Hint: use the fact that the solutions, " n x ( ) , are eigenfunctions of the Hamiltonian, and the fact that they are orthonormal. (c) Evaluate the expectation value of position x to show that it is x t = L 2 16 L 9 π 2 cos E 2 E 1 h t So the mean position oscillates around the middle of the box at L/2. From this equation sketch a plot of x t as a function of time. 3. The Hamiltonian for a particle of mass M constrained to move on a ring is H = " ! 2 2 I # 2 #$ 2 and the Schrödinger equation is H " # ( ) = E " # ( ) (1) where I = M a 2 is called the moment of inertia, M is the mass of the particle, and a is the radius of the ring. " is the angular coordinate that specifies the location of the particle on the ring and thus lies on the interval 0 " # " 2 $ . (a) By substituting in to the Schrödinger equation show that the solutions to equation (1) are " # ( ) = A e im # where m = 2 I E ! 2 . " a
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