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Week5 2023

The document contains problems related to quantum mechanics, specifically focusing on matrix quantum mechanics and the simple harmonic oscillator. It includes questions on Hamiltonians, eigenvalues, expectation values, and operator formalism. The problems require calculations and proofs involving quantum states and operators in a two-state system and the simple harmonic oscillator framework.

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namkoi00
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11 views1 page

Week5 2023

The document contains problems related to quantum mechanics, specifically focusing on matrix quantum mechanics and the simple harmonic oscillator. It includes questions on Hamiltonians, eigenvalues, expectation values, and operator formalism. The problems require calculations and proofs involving quantum states and operators in a two-state system and the simple harmonic oscillator framework.

Uploaded by

namkoi00
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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QUANTUM MECHANICS B SPA6413 PROBLEMS WEEK 5

ALL questions will be marked.

QUESTION 1:! Matrix quantum mechanics: The Hamiltonian of a!two-state system is given by
E1 0 2 −1
H= and another operator is given by A = .
0 E2 −1 2
(a) Write down the energy
!
eigenvalues, and the energy eigenvectors U(1) and U(2) of H.
1
Show that V = √12 is a normalised eigenvector of A. Find the corresponding eigenvalue. [3]
1
(b) The state of the system at t = 0 is C(0) = V . Find the expectation values hHi and hH 2 i. [2]

(c) If the state at t = 0 is C(0) = V find the state C(t) at later times t > 0 using the expansion
theorem. Hint: write V as a linear combination of U(1) and U(2) and put ”wiggle factors”. [2]

(d) Calculate the expectation values hAi and hA2 i using the time dependent state C(t). Hence,
find the uncertainty ∆A. [4]

Hint: recall the expectation values of operators in matrix quantum mechanics are defined as
hOi = C † OC where O is the matrix form of the operator and C is the state; either C(0) or C(t)
in the examples above.

QUESTION 2: Simple Harmonic Oscillator (SHO):


Summary of important formulas of the 1D √ simple harmonic oscillator
√ (SHO): R
† 1 † † ∗
Ĥ = h̄ω(â â + 2 ), [â, â ] = 1, â ψn = n + 1ψn+1 , âψn = nψn−1 , hm|ni = ψm ψn dx = δmn
(orthonormality), where ψn are energy eigenfunctions and the corresponding ket is defined q as |ni.
† −ih̄β † †
x̂ = 2β (â + â ), p̂x = 2 (â − â ), â = 2 (βx + i βh̄ p̂x ), â = 2 (βx − i βh̄ p̂x ) with β = mω
√1 √ √1 1 √1 1
h̄
.

(a) Using the expressions for â and ↠in terms of x̂ and p̂x given above and the known commutator
between x̂ and p̂x , prove the commutation relation [â, ↠] = 1. [2]
2 2
(b) The energy eigenstate ψ1 ∝ xe−β x /2 up to normalisation. Ignoring the normalisation con-
stant find ψ2 by acting with the raising operator ↠on ψ1 . Note: here you need to write the raising
∂
operator as a differential operator using that p̂x = −ih̄ ∂x . [2]

(c) In the following you must use the operator formalism i.e. express operators in terms of â and
↠and freely use all relations given above. The matrix representing an operator Ab has mn-th
R ∗ b
element Amn = ψm Aψn dx = hm|A|ni b , where m, n = 0, 1, 2, . . .
−ih̄β
Obtain the matrix form of the operators p̂x = √ (â
2
− ↠), and write out explicitly the first
4 × 4 part of this matrix. [5]

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