p827aps7 - a = i A a a . (4) Solve this equation for A a a...

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Physics 827: Problem Set 7 Due Wednesday, November 14, 2007 1. (30 pts.) Coherent Quasi-Classical States of a Harmonic Oscil- lator. Consider a harmonic oscillator described by the Hamiltonian H = ( a a + 1 2 , where ω = r k/m , k is the spring constant and m is the mass of a harmonic oscillator, and a and a are the annihilation and creation operators as described in Shankar. A coherent state is deFned to be an eigenstate | α a of the annihilation operator a with eigenvalue α , i. e., a | α a = α | α a . (1) Since a is not Hermitian, the eigenvalues α need not be real. (a). Consider the state | α a = exp( −| α | 2 / 2) s n =0 α n n ! | n a , (2) where | n a is an eigenstate of H with energy ( n + 1 2 , Show that this state is a normalized eigenstate of a with eigenvalue α . You will need various properties of the creation and annihilation operators as given in Shankar. (b). The expectation value A α | a | α a ≡ A a a satisFes the equation of motion d dt A a a = i ¯ h A [ a, H ] a . (3) [This is a special case of Shankar, eq. (6.2)]. By evaluating the right- hand side, show that d dt A a
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Unformatted text preview: a = i A a a . (4) Solve this equation for A a a ( t ) in terms of A a a (0). 1 (c). According to Shankar, eq. (7.4.3), a a A = r m/ (2 h ) a X A + i r 1 / (2 m h ) a P A . Now, consider the classical harmonic oscillator, whose equation of mo-tion is just x = ( k/m ) x, (5) where k/m = 2 . By solving for the classical x ( t ) and p ( t ) , show that a X A ( t ) and a P A ( t ) reduce to the classical solutions if the quantum-mechanical wave function is the coherent-state wave function deFned in (a). This is why the coherent state is sometimes called the quasi-classical state. (d). Not to be turned in. Show that in the coherent state, the expec-tation value of the commutator X P is the minimum value allowed by the Heisenberg uncertainty principle. 2. (10pts.) Shankar, exercise 10.2.2. 3. (10pts.) Shankar, exercise 10.2.3. 2...
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p827aps7 - a = i A a a . (4) Solve this equation for A a a...

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