PH-107 (2020): Tutorial Sheet

1. Consider an asymmetric nite potential well of width L, with a barrier V1 on one side and

a barrier V2 on the other side (gure below). Obtain the energy quantization condition for the bound states in such a well. From this condition derive the energy quantization

conditions for a semi-innite potential well (when V1 → ∞ and V2 is nite)

2. A particle of mass m is bound in a double well potential shown in the gure above. Its

energy eigen state ψ(x) has energy eigenvalue E = V0, where V0 is the energy of the plateau in the middle of the potential well. It is known that ψ(x) = C, a constant in the plateau region.

(a) Obtain ψ(x) in the regions −2L < x < −L and L < x < 2L and the relation between the wave number k and L. (b) Determine C in terms of L. (c) Assume the bound particle to be an electron and L = 1 Å. Calculate the two lowest

values of V0 (in eV) for which such a solution exists. (d) For the smallest allowed k, calculate the expectation values for x, x2, p and p2 and show that the uncertainty principle is obeyed.

3. You are given an arbitrary potential V (x) and the corresponding orthogonal and nor-

malized bound-state solutions to the timeindependent Schrodinger's equation, ϕn(x) with

the ocrresponding energy eigen values En. At time t = 0, the system is in the state,

ψ(x, 0) = A[ϕ1(x) + ϕ2(x) + ϕ4(x)].

(a) Find the value of A. (b) What is the wave function at timet > 0. (c) What is the expectation value of the energy at time t > 0.

1 4. A nite square well (height = 30 eV, width 2a from −a to +a) has six bound states 3, 7, 12, 17, 21, and 24 eV. If instead, the potential is semiinnite, with an innite wall at x = 0, how many bound states will exist and what are the energies associated with it? Justify your answer.

5. A particle with energy E is bound in a nite square well potential with height U and width 2L (from −L to +L).

(a) If E < U, obtain the energy quantization condition for the symmetric wave functions p p in terms of K and α, where K = 2mE/~2 and α = 2m(U − E)/~2. (b) Apply this result to an electron trapped inside a defect site in a crystal. Modeling this defect as a nite square well potential with height 5 eV and width 200 pm, calculate the ground state energy (c) Calculate the total number of bound states with symmetric wave-function

6. A potential V (x) is dened over a region R, which consists of two sub regions R1 and R2

(R = R1 ∪ R2). This potential has two normalized energy eigenfunctions Ψ1(x) and Ψ2(x)

with energy eigenvalues E1 and E2 (E1 6= E2), respectively. Ψ1(x) = 0 outside the region

R1 and Ψ2(x) = 0 outside the region R2.

(a) Suppose the regions R1 and R2 do not overlap, show that the particle will stay there

forever, if it is in the region R1. √ (b) If the initial state is Ψ(x, 0) = [Ψ1(x)+Ψ2(x)]/ 2, show that the probability density |Ψ(x, t)|2 is independent of time.

2 (c) If the regions R1 and R2 overlap, show that the probability density |Ψ(x, t)| oscillate in time for the initial state given in (b).\