p827aps7 - a = − iω A a a(4 Solve this equation for A a...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: a = − iω 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 mω/ (2¯ h ) a X A + i r 1 / (2 mω ¯ 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 = ω 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 Δ X Δ 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...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

p827aps7 - a = − iω A a a(4 Solve this equation for A a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online