Quantum Mechanics | Week 1

Quantum Mechanics Week 1 Answers

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Q1. The energy eigenvalue and the corresponding eigenfunction for a particle of mass m in a one-dimensional potential V(x) are,
E=0, (x)=A/(x^2+a^2)
where A is a positive constant. The form of potential V(x) is
a) h2(3x^2-a^2)/m(x^2+a^2)^2
b) h2(x^2-a^2)/2m(x^2+a^2)^2
c) h2(3x^2-a^2)/m(x^2+a^2)^3
d) h2(x^2-a^2)/2m(x^2+a^2)^3

Q2. Consider a photon that scatters from an electron at rest. If the photon scatters at an angle 60° with a wavelength (λ’) that is twice its incident wavelength (λ), then the angle at which the electron recoils will be _________ degrees. (Answer should be an integer.)

Answer: 30

Q3. For a particle in a one dimensional box, the ground state energy is given by E₀. The allowed energies of such a particle are
a) 2E₀
b) 4E₀
c) 6E₀
d) 9E₀

Q4. Consider a particle in a two dimensional box whose ground state energy is €₀. Then the energy E = 13€₀ is two fold degenerate.
a) True
b) False

Q5. Suppose the 2nd energy level of a particle in a cubical box is given by E = 6E_g where E_g, is the ground state energy. Then the immediate 3rd energy level will be ______ fold degenerate. (Answer should be an integer.)

Answer: 3

Q6. The non-zero commutator brackets are
a) [z, p_x]
b) [z,p_y]
c) [x^2, p_x]
d) [x, p_x²]

Q7.

Answer: c

Q8.

Answer: d

Q9. The normalized wavefunction of a particle is ¥(x) = Aexp(iax – ibt), where A, a and b are constants. The uncertainty in its momentum is
a) ha
b) h²a’²
c) 0
d) -ha

Q10.

Answer: True