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: Wavefunction for Particle, Dependance on Position, Valid Wavefunction, Allowable Wavefunction, Energy Eigenfunctions, Probability of Observing Particle, Expectation Value of Energy, Time-Scale Associated
Typology: Exams
1 / 8
This page cannot be seen from the preview
Don't miss anything!
Level 2 (PRINCIPAL COURSE)
Friday 18 January, 13:00 - 15:
PHYSICS/ASTROPHYSICS
PHY-
QUANTUM MECHANICS
Candidates should attempt to answer FOUR questions.
An information sheet is provided on the last page of this exam.
Tables of physical and mathematical data may be obtained from the invigilator.
i. state why the function is not allowable as part of a wavefunction; ii. state all values of x at which the function is not valid.
f 1 (x) =
{ e+x^ (x < 0) e−x^ (x ≥ 0) f^2 (x) =
{ sin(πx) (0 < x < 1) π/x (x ≥ 1)
f 3 (x) = (^) ±^1 √x (x ≥ 0) f 4 (x) = tan x (−π/ 2 ≤ x ≤ π/2) [40] (b) The wavefunction for a particle has the following dependance on position, x.
ψ(x) =
{ A sin(x)/x x < π/ 2 Be−βx^ x ≥ π/ 2
Find the value of β for which the wavefunction satisfies the requirements for an allowable wavefunction at x = π/2. Explain your method clearly. [45] (c) Explain why the derivative ∂ ∂tΨ must be finite for a valid wavefunction, Ψ. [15]
V = ∞ for x ≤ 0 V = 0 for 0 < x < a V = ∞ for x ≥ a.
The particle is in a state described by the eigenfunction
ψ 3 (x) =
√ 2 a
sin(3πx/a).
(a) What is the Born interpretation of the wavefunction? [10] (b) Sketch the probability distribution for the observed position of the particle with the eigenfunction ψ 3 (x). [20] (c) Show that ψ 3 (x) is a solution of the 1-dimensional time-independent Schr¨odinger equation for this potential in the region (0 < x < a) and hence derive an ex- pression for the energy of this state in terms of a and m. [40] (d) Explain briefly why the expectation value for the position must have the value 〈x〉 = 12 a. [10] (e) Discuss briefly whether the following two statements are consistent with each other.
V (x, y, z) =
mω^2 (x^2 + y^2 + z^2 )
where m is the mass of particle. The solutions of the energy eigenvalue equation for a harmonic oscillator in 1 di- mension are ψn with energies
En =
( n +
) ¯hω, n = 0, 1 , 2 ,....
(a) Explain what is meant by the statement “ψn is a solution of the energy eigen- value equation with energy En”. [10] (b) Show that ψnx,ny ,nz (x, y, z) = ψnx (x)ψny (y)ψnz (z) is a solution of the 3-D time independent Schr¨odinger equation for a harmonic oscillator if ψnx (x), ψny (y) and ψnz (z) are solutions of the 1-D time indepen- dent Schr¨odinger equation for a harmonic oscillator on the x-, y- and z-axes, respectively. Hence, show that the energy of the state ψnx,ny ,nz for the 3-D harmonic oscillator is
Enx,ny ,nz =
( nx + ny + nz +
) ¯hω.
(c) Write down the energies of the three lowest energy states and give the degen- eracy of these three energy levels. [30] (d) What is the value of the commutator [ˆx, yˆ]? What is the implication of this value for the observed values of x and y? [20]
3
0
E V
8 8
B
x
V(x)
−3 −2 −1 1 2
(a) What is the value of the wavefunction for the particle for x < −3. Explain your answer. [10] (b) Sketch the shape of the eigenfunction in the region (− 4 < x < 4) for a particle in the ground state. Indicate the type of mathematical function you have used for each region of x on your sketch (e.g., “exponential”). [30] (c) What is the parity of the ground state? Justify your answer. [15] (d) A particle with energy E < VB is placed in the left-hand side of the potential. Compare the location of the particle when observed at later times predicted by classical physics and by quantum phsyics. [20] (e) Sketch the eigenfunction in the case where the potential at x > 3 and x < − 3 is V (x) = 2VB rather than V (x) = ∞. Indicate the type of mathematical function you have used for each region of x on your sketch. [25]
One dimensional form
−
¯h^2 2 m
∂x^2
∂t Three dimensionsal form
i¯h
∂t
[ −
¯h^2 2 m
∇^2 + V (r)
] Ψ
Time-independent Schr¨odinger equation in 1-dimension
¯h^2 2 m
d^2 ψ dx^2
Time-independent Schr¨odinger equation in 3-dimensional Cartesian coordinates.
¯h^2 2 m
( ∂^2 ∂x^2
∂y^2
∂z^2
) ψ + V (x, y, z) ψ = Eψ
