Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Past Exam of Quantum Mechanics which includes Wavefunction for Particle, Valid Wavefunction, Stern Gehrlach Device, Square Barrier, Spin System etc. Key important points are: Physical Interpretation, Wave Function, Observed Position of Particle, Normalised Wavefunction, Quantisation of Energy, Wave-Particle Duality, Expectation Value for Position, Oscillator Potential
Typology: Exams
1 / 8
This page cannot be seen from the preview
Don't miss anything!
Level II
Thursday 12th^ January 2012, 09.30 – 11:
PHYSICS/ASTROPHYSICS
PHY-
QUANTUM MECHANICS
Candidates should attempt ALL of PART A and TWO questions from PART B.
PART A yields 40% of the marks, PART B yields 60%.
A sheet of useful formulae can be found on page 8.
PART A Answer all TEN questions
A1 Give a physical interpretation of the wave function Ψ in terms of the observed position of the particle and explain how this leads to the concept of a normalised wavefunction. [4]
A2 Explain the following concepts and give an example of a physical system that demonstrates each concept.
A3 Calculate the expectation value for the position, 〈x〉, of a particle with the normalized wave function Ψ(x, t) =
2 π e−πx^ e−iωt^ x > 0. You may use the following integral without proof in your answer. ∫ (^) ∞ 0 r
k (^) exp(−αr) dr = k! αk+ [4]
A4 State three differences between the predictions of classical physics and the predictions of quantum mechanics for the properties of a particle in a harmonic oscillator potential, V (x) = 12 kx^2. [4]
A5 Sketch the energy eigenfunction, ψ 1 and ψ 2 , for the ground state and first excited state of the finite square-well potential
V (x) =
0 x < −a −V 0 −a ≤ x ≤ a 0 x > a
PART B Answer TWO out of FOUR questions
B1 A particle of mass m is trapped in the following potential:
V (x) =
∞ x < 0 0 0 ≤ x ≤ a ∞ x > a (a) Show that the solutions of the time independent Schr¨odinger equation are
ψn(x) =
(^0) √ x < 0 (^2) a sin(nπx/a) n = 1, 2 , 3 ,... , 0 ≤ x ≤ a 0 x > a and so derive an expression for the energy of the particle in terms of a and m. [20] (b) What is the expectation value for x in this case? Justify your answer. (Hint: No calculation required.) [3] (c) Write down the wavefunction, Ψ(x, t), for this particle in terms of a and m [3] (d) Discuss briefly whether the following two statements are consis- tent with each other.
/Cont’d
B2 Consider a particle with the wave function Ψ(x, t) = ψ(x)e−iωt^ where
ψ(x) =
0 x ≤ − 1 A [1 + cos(πx)] − 1 < x < 1 0 x ≥ 1 (a) Normalize this wavefunction. [8] (b) Show that the ground state has definite parity and state its value. [4] (c) Calculate the uncertainty in the observed position, ∆x. [8] (d) Show that ψ has the required mathematical properties for a valid wave function at the boundaries x = ±1. [5] (e) Discuss whether Ψ(x, t) can be a valid wave function for a par- ticle in a harmonic oscillator potential if it is not one of the energy eigenfunctions, ψn(x). [5]
You may use the following standards integrals without proof in your answers. ∫ [1 + cos(x)]^2 dx =^14 [6x + 8 sin(x) + sin(2x)] + C
∫ (^1) − 1 x
(^2) [1 + cos(πx)] (^2) dx = 1 − 15 2 π^2
/Cont’d
B4 The wave functions for an electron in a simple model of the hydrogen atom have the form Ψn,ℓ,mℓ (r, θ, φ, t) = u( rr )Yℓ,mℓ (θ, φ)e−iEt/¯h.
The radial eigenfunction for an electron in a hydrogen atom in the 2p state is u(r) = √ 241 a 0
( (^) r a 0
) 2 e−r/^2 a^0 , where a 0 is the Bohr radius.
(a) State the physical quantity most closely associated with each of the quantum numbers n, ℓ and mℓ, and state the possible values for each quantum number for an electron in a 2p state. [6] (b) Calculate the expectation value, 〈r〉, for an electron in the 2p state. You may use the following standard integral without proof in your answer. ∫ (^) ∞ 0 r
k (^) exp(−αr) dr = k! αk+ [8] (c) With the aid of a sketch, explain why 〈r〉 is different from the most probable observed value of r for the electron. [4] (d) With the aid of a labelled diagram, describe the main features of the Stern-Gerlach experiment. Explain how this experiment shows that the eigenfunctions Ψn,ℓ,mℓ do not give a complete description for the properties of an electron in a hydrogen atom. [12]
/Cont’d
Quantum Mechanics formulae
Time independent Schr¨odinger equation
d^2 ψ dx^2 +
2 m ¯h^2 [E^ −^ V^ (x)]^ ψ^ = 0
