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Unformatted text preview: PHY4221: Homework 5 Due Oct. 9, 2008 1. In the 1D inﬁnite square well problem deﬁned in the region |x| ≤ L, an extra potential
V = kx is added. Express the Hamiltonian of the system as a form of matrix in the basis
deﬁned by the eigenvectors of the original square well Hamiltonian.
2. The Hamiltonian of a system is given by
H = αp2 + βx,
where α and β are constants. Find and solve the Heisenberg equation of motion of the
position and momentum operators. The initial state is a Gaussian wave packet, centered
at zero speed and x = 0, with a position width given by σ . Find expectation value of the
position and its variance as a function of time.
3. (a) Show that
= + B + B B + B B B + .....
A AA AAA AAAA
where A and B are two operators whose inverse exist. Then calculate the expectation value
of 1/(A − B ) with respect to the state |0 , given that B |0 = |1 and B |1 = |0 , and |0
and |1 are eigenvectors of A with eigenvalues λ1 and λ2 .
(b) Investigate the accuracy in the power of τ in the following expression:
e−i(A+B )τ ≈ e−iAτ /2 e−iBτ e−iAτ /2
where τ is a small number.
(c) Show that [A, B ] = iqI , where q is a real number and I is the identity matrix, cannot
be satisﬁed by any ﬁnite dimensional Hermitian matrix A and B .
4. Let A1 , A2 and A3 be three operators satisfying the commutation relations: [A1 , A2 ] =
iA3 , [A2 , A3 ] = iA1 , [A3 , A1 ] = iA2 . Consider the Hamiltonian, H = ΩA3 , ﬁnd and solve the
Heisenberg equation of motion of A1 (t) and A2 (t). Then from your solutions, ﬁnd the rule
of transformation for eiθA3 A1 e−iθA3 . 1 ...
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.
- Spring '11