The second quiz for PHYs-2351-FR01A: Quantum Physics
Name: ________________
ID: ____________________
Part A: Multiple Choices (each correct answer is worth 5)
1. A particle in an infinite well in the n=2 stationary state is most likely to be found
a). in the center of the well;
b). one-third of the way from either end;
c). one-quarter of the way from either end;
d). it is equally likely to be found at any point in the well.
2. In the following plots, which is the one describes the energy level of infinite well
a)
c)
b)
d)
3. Unlike the infinite well, the finite well can only hold finite number of bound states. And we
need to solve a transcendental equation which cannot be solved by pencil-and-paper
techniques. Therefore, we figure out graphing is a good way to see the energy quantization
within the finite well. A graph like this is given below. How many bound states contained in this
2𝑚𝐸
?
ℏ2
finite well with height equal to 𝑈0 , width equals to 𝐿 and 𝑘 is defined as 𝑘 ≡ √
a)
b)
c)
d)
1
2
3
4
4. There is an infinite well whose potential is described by
0 |𝑥| < 𝑎
𝑉(𝑥) = {
∞ |𝑥| > 𝑎
This potential can also be plotted as
ℏ2 𝑑 2 𝜓
The Schrödinger equation within the well can be written as − 2𝑚 𝑑𝑥 2 = 𝑘 2 𝜓, where 𝑘 2 =
2𝑚𝐸
.
ℏ2
The general solution of this function is 𝜓(𝑥) = 𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥. To decide the coefficient A
and B, we need to make use of the boundary condition. What can we obtain at 𝑥 = 𝑎?
a) −𝐴 sin 𝑘𝑎 + 𝐵 cos 𝑘𝑎 = 0;
b) 𝐴 sin 𝑘𝑎 − 𝐵 cos 𝑘𝑎 = 0;
c) 𝐴 sin 𝑘𝑎 + 𝐵 cos 𝑘𝑎 = 0;
d) 𝐴 sin 𝑘𝑎 + 𝐵 cos 𝑘𝑎 = 1
5. Which of the following expression is used to define Hermiticity of a given operator 𝐴̂?
∗
a) ∫ 𝜓 ∗ 𝐴̂𝜓𝑑𝑥 = ∫(𝐴̂𝜓) 𝜓𝑑𝑥
b) 𝐴̅ = ∫ 𝜓 ∗ 𝐴̂𝜓𝑑𝑥
c) ∫ 𝜓 ∗ 𝐴̂𝜓𝑑𝑥 = ∫ 𝜓𝐴̂𝜓 ∗ 𝑑𝑥
d) 𝐴̅ = ∫ 𝜓𝐴̂𝜓𝑑𝑥
6. For bound states, we have so-called boundary conditions to figure out the acceptable
solutions 𝜓(𝑥). Which of the following statements is NOT a reasonable boundary condition?
a) 𝜓(𝑥) is a continuous function;
b) 𝜓(𝑥) = 0 if 𝑥 is in a region where it is physically impossible for the particle to be;
c) 𝜓(𝑥) → 0 as 𝑥 → +∞ and 𝑥 → −∞
d) 𝜓(𝑥) is a normalized function;
e) 𝜓(𝑥) = 0 at the edge of two neighboring regions.
ℏ
7. Which of the following statement about uncertainty principle Δ𝑥Δ𝑝 ≥ 2 is NOT correct?
a) This principle means no matter how clever we are, and no matter how good our experiment
is, we cannot measure both 𝑥 and 𝑝 simultaneously with arbitrarily good precision;
b) This principle is a reflection of duality of matter wave;
c) In no condition we can reach Δ𝑝 = 0;
d) This principle can be used in estimation to decide qualitative property of some quantum
systems.
8. The first three energy levels and the related wave functions of the simple harmonic oscillator
is given by the figure below
Based on the information provided by the graph above, and the relation probability density =
|𝜓(𝑥)|2 , which probability density in the graph below represents a quantum harmonic oscillator
5
with 𝐸 = 2 ℏ𝜔? 𝜓(𝑥) is the wave function.
9. After performing variables separation on a time-dependent Schrödinger equation, we get the
“temporal part” as
𝑖ℏ
1 𝑑𝜙(𝑡)
=𝐶
𝜙(𝑡) 𝑑𝑡
Which of the following is the possible solution to this equation?
𝐶
a) 𝜙(𝑡) = 𝐶𝑡 2 b) 𝜙(𝑡) = 𝑒 −𝑖(𝐶/ℏ)𝑡 c) 𝜙(𝑥) = sin ℏ 𝑡 d) 𝜙(𝑥) = 𝐶 2
10. According to Born’s interpretation of the wave function, |Ψ(𝑥, 𝑡)|2 is
a) probability
b) probability density
c) energy
d) force
Part B. Short Answer Questions (each of them is worth 10)
11. An electron is in a state given by the wave function
2
𝜓(𝑥) = 𝐴𝑒 −[(𝑥−𝑏)/2𝜀]
a) the value of 𝐴
b) the expectation value of the position. (hint: the value can be obtained by calculating 𝑥̅ =
∫all space 𝑥|𝜓(𝑥)|2 𝑑𝑥)
(The Gaussian integrals needed here are
∞
𝜋
2
∫ 𝑒 −𝑎(𝑧−𝑏) 𝑑𝑧 = √
𝑎
−∞
and
∞
2
∫ 𝑧𝑒 −𝑎(𝑧−𝑏) 𝑑𝑧 = 𝑏√
−∞
)
𝜋
𝑎
12. Air is mostly 𝑁2 , diatomic nitrogen, with an effective spring constant of 𝜅 = 2.3 × 102 N/m,
and an effective oscillating mass of half the atomic mass, which is 𝑚 = 1.16 × 10−26 kg. For
roughly what temperature should vibration contribute to its heat capacity? (ℏ =
𝜅
1.055 × 10−34 J ⋅ s, and Δ𝐸 = ℏ𝜔0 with 𝜔0 = √𝑚)
13. If Ψ1 and Ψ2 are two solutions of a time-independent Schrödinger equation
−
ℏ2 𝑑 2 Ψ
+ 𝑈(𝑥)Ψ = 𝐸Ψ,
2𝑚 𝑑𝑥 2
Show that 𝑎1 Ψ1 + 𝑎2 Ψ2 is also a solution of this equation, where 𝑎1 and 𝑎2 are two arbitrary
constants.
14. We have discussed the infinite well like the one in the plot below
We figured out the wave function within the well can be expressed as
2
𝑛𝜋𝑥
𝜓𝑛 (𝑥) = √ sin
𝐿
𝐿
𝑛 = 1, 2, 3, …
𝐿
Show that the average value of 𝑥 is 2, independent of the quantum state.
The integral you may need:
∫ 𝑥 sin2 (
𝑛𝜋𝑥
𝑥2
𝐿𝑥
2𝑛𝜋𝑥
𝐿2
2𝑛𝜋𝑥
) 𝑑𝑥 =
−
sin (
) − 2 2 cos (
)
𝐿
4 4𝑛𝜋
𝐿
8𝑛 𝜋
𝐿
15. In Young’s double-slit experiment. Let Ψ1 = |Ψ1 |𝑒 𝑖𝛼1 and Ψ2 = |Ψ2 |𝑒 𝑖𝛼2 are the two waves
pass through the two slits, where 𝛼1 and 𝛼2 are two phases. Let 𝐼1 = |Ψ1 |2 and 𝐼2 = |Ψ2 |2
denote the intensity of incoming waves, respectively. Verify the interference term is
2√𝐼1 𝐼2 cos(𝛼1 − 𝛼2 ). (Hint: Calculate 𝐼 = (Ψ1 + Ψ2 )∗ (Ψ1 + Ψ2 ))