Assignment – 2
Date: 4 August 2026
Instructions. Answer all questions and show the essential steps in numerical and derivation problems.
For a question that asks you to select all correct statements, one or more options may be correct. Unless
stated otherwise, a one-dimensional infinite box occupies 0 < x < L, and a cubic box has side length L.
1. State the first postulate of quantum mechanics in terms of a state vector or wavefunction. Explain
why Ψ and eiθ Ψ, where θ is a real constant, describe the same physical state.
2. Distinguish between the wavefunction Ψ(x, t) and the probability density |Ψ(x, t)|2 . State their
physical dimensions in (i) one dimension and (ii) three dimensions.
3. Let |1⟩ and |2⟩ be orthonormal. Which state is normalized?
(A) |1⟩ + i|2⟩
1
(B) √ |1⟩ + i|2⟩
2
1
(C)
|1⟩ + i|2⟩
2
√
(D) 2 |1⟩ + i|2⟩
4. In a region where the potential is finite and nonsingular, select all correct statements. An acceptable
wavefunction should be
(A) single-valued,
(B) finite and continuous,
(C) square-integrable,
(D) continuously differentiable wherever the potential remains finite and nonsingular.
5. For 0 < x < L, let
ψ(x) = A x(L − x),
and let ψ(x) = 0 outside the box. If the wavefunction is normalized, find the numerical value of
AL5/2 .
6. For a continuous position coordinate, the probability of finding a particle at exactly one specified
point x = x0 is
(A) |ψ(x0 )|2 ,
(B) zero, although the probability density at x0 may be nonzero,
(C) one for every normalized state,
(D) undefined in quantum mechanics.
7. A particle has the normalized probability density
p(x) =
3x2
,
L3
0 < x < L,
and p(x) = 0 outside this interval. Find the probability of locating the particle in 0 < x < L/2.
8. Sketch a real wavefunction that changes sign once. On the same horizontal scale, sketch its
probability density. Mark the node and explain which information is lost when ψ is replaced by
|ψ|2 .
9. Write the one-dimensional time-dependent Schrödinger equation for a particle of mass m in a
potential V (x), and identify every term in the equation.
10. Write the corresponding time-independent Schrödinger equation. Under what condition on the
potential is separation into a spatial part and a temporal part possible in the usual stationary-state
form?
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11. Substitute
Ψ(x, t) = ψ(x)T (t)
into the time-dependent Schrödinger equation for a time-independent potential. Separate the
variables and show that both sides must equal the same constant E.
12. The temporal factor associated with energy E is
(A) e−Et/ℏ
(B) e+Et/ℏ
(C) e−iEt/ℏ
2
(D) e+iEt /ℏ
13. At time t = h/(4E), the temporal factor e−iEt/ℏ equals
(A) 1
(B) −1
(C) i
(D) −i
14. For
Ψ(x, t) = ψ(x)e−iEt/ℏ ,
show directly that the probability density is independent of time.
15. Select all correct statements about a single stationary state.
(A) The full wavefunction generally depends on time.
(B) Its probability density is independent of time.
(C) Its temporal factor has unit magnitude.
(D) Its spatial part must always be a real-valued function.
16. Consider the following assertion and reason.
Assertion: The probability density of a single stationary state is independent of time.
Reason: The factor e−iEt/ℏ has modulus one.
Choose the correct option:
(A) Both the assertion and the reason are true, and the reason correctly explains the assertion.
(B) Both the assertion and the reason are true, but the reason does not correctly explain the
assertion.
(C) The assertion is true, but the reason is false.
(D) The assertion is false, but the reason is true.
17. Define the phase period τ by
e−iE(t+τ )/ℏ = e−iEt/ℏ .
If E2 = 4E1 , then
(A) τ2 = 4τ1
(B) τ2 = 2τ1
(C) τ2 = τ1
(D) τ2 = τ1 /4
18. Match each expression to its description:
iℏ ∂Ψ/∂t
ℏ2 d2 ψ
II. −
+Vψ
2m dx2
III.
e−iEt/ℏ
IV.
|Ψ(x, t)|2
I.
(a) temporal factor,
(b) right side of the TDSE,
(c) spatial side of the TISE,
(d) probability density.
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Write the matching as I–?, II–?, III–?, IV–?.
19. Distinguish a general time-dependent wavefunction from a stationary-state wavefunction. In
particular, explain what is stationary and what still changes with time.
20. Why is n = 0 not an allowed quantum number for a particle in a one-dimensional infinite box?
(A) It would give an infinitely large energy.
(B) It would make the spatial wavefunction identically zero.
(C) It would violate normalization because the wavefunction becomes infinite.
(D) It would place a node outside the box.
21. Verify that
r
2
nπx
ψn (x) =
sin
L
L
is normalized on 0 < x < L for every positive integer n.
22. The number of nodes strictly inside the box for the state n = 5 is
(A) 3
(B) 4
(C) 5
(D) 6
23. For the state n = 2, select all correct statements.
(A) The probability density is zero at x = 0, L/2, L.
(B) The probability density is maximum at x = L/4 and x = 3L/4.
(C) The wavefunction has one interior node.
(D) The probability in the left lobe is different from that in the right lobe.
24. For any stationary state of the box, the probability of locating the particle in the left half,
0 < x < L/2, is
(A) 0
(B) 1/4
(C) 1/2
(D) dependent on whether n is odd or even.
25. Find
E3 − E1
.
E1
26. If the box length changes from L to 2L, while m and n remain fixed, the energy becomes
(A) 2En
(B) 4En
(C) En /2
(D) En /4
27. The box length is increased by 0.10%. Using the first-order fractional change, determine the
percentage change in any fixed-n energy level. Include the sign.
28. For a fixed stationary state, match each change to the corresponding energy scaling:
I.
m → 2m
II.
L → 3L
III.
n → 2n
IV. no parameter changed
Write the matching as I–?, II–?, III–?, IV–?.
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(a) E → 4E,
(b) E → E/2,
(c) E → E,
(d) E → E/9.
29. If ψnx ny nz (x, y, z) is the spatial part of a stationary state with energy Enx ny nz , write the corresponding full time-dependent wavefunction.
30. The ground state of a cubic infinite box is labelled by
(A) (0, 0, 0) and has zero energy,
(B) (1, 1, 1) and has nonzero energy,
(C) (1, 0, 0) and is triply degenerate,
(D) (2, 2, 2) and is nondegenerate.
31. In a cubic box, find the ratio of the energy of the state (2, 2, 1) to the ground-state energy.
32. Find the degeneracy of the first excited energy level of a cubic box.
33. In a cubic box, the level with
n2x + n2y + n2z = 14
has degeneracy
(A) 1
(B) 3
(C) 4
(D) 6
34. A cubic-box energy level is four times the ground-state energy. Find its degeneracy.
35. If Lx , Ly , Lz are all unequal, which statement is generally correct?
(A) Every permutation of (nx , ny , nz ) has the same energy.
(B) Permutation degeneracy is generally removed.
(C) The ground-state energy becomes zero.
(D) Every excited level becomes threefold degenerate.
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