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Unformatted text preview: Chem 120A Hour Examination #1 1. Consider a free particle in one dimension, :17. Its momentum operator is .. h d
p _ 2 dx '
6 MS a. Show that both
d”; : eikz 7 k > 0) (Z)
lib—k : 6—21.11. ) k > 07 (M)
are eigenfunctions of p.
6 pts b. For functions (2) and (22), what are the expectation values of p in terms of
fundamental constants and k.
6 pts 0. Is
«b cc 2/2}; + 1/24. (22:21)
an eigenfunction of p?
6 PtS d. In terms of fundamental constants and Is, what is the expectation value of p
for wave function (222).
6 Pts e. In terms of fundamental constants and k, what are the expectation values of p2 for each of wave functions (2), (22) and (222). 2. Imagine a Hilbert space spanned by only two functions, 1/11 and $2, where
and Here, H is the Hamiltonian operator and Ej is the energy associated with the
normalized wave function 212]; Accordingly, the 1b j’s are the stationary solutions to the general time dependent Schrodinger equation, . a
ma—tQ/(t) 2 HM) . 1 (For notational simplicity, functional dependence on spatial coordinates are su— pressed in this formula.) Suppose there is some other operator, L, for which [xi/)1 = ”W2 and L1/J2 = “M7751 10 P135 a. Is L2 a constant of the motion? That is, for an arbitrary \I’(t), is its expec- tation value, (L2), independent of time? Explain. 10 PtS b. Evaluate the expectation values of i. L ii. L2 and iii. (L — (L))2 for the stationary state 1&1 and for the stationary state 1/22.
10 PtS c. Evaluate (¢1|L|1/22) and (1,1)2ILW1) and show L is Hermitian. Suppose that at time t = 0, the wave function is W0) = 53% + M2) 5 Pts d. In terms of 1/)1, 1/)2, E1, E2 and fundamental constants, what is \I/(t) for all subsequent times? 15 pts e. Evaluate as a function of time the expectation value of L, (II/(t)lLl\Il(t)) Express your result in terms of E1, E2, t and fundamental constants. 3. A particle of mass m exists in the potential V(a:) drawn in the ﬁgure V (x) 5 pts 5 pts 5 pts 5 pts The energy E is the lowest stationary state energy. . Will a particle of mass m', where m’ > m, in the same potential have a ground state energy lower than E? . Will the ground state wave function be zero at the classical turning points a and b? . In the region a < :r < b, how many nodes will the ground state wave function have? . The principal spectral line for absorption is due to transitions from the ground state to the ﬁrst excited state. Let 1/ denote the frequency of the
absorbed light. Consider 1/ as a function of the particle mass, m, and sup-
pose l/(m) 0: mp What is the power p? ...
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- Spring '10
- pts, fundamental constants