UNIT II: QUANTUM MECHANICS
Short Answer Type Questions
1. Define wave function Ψ
The quantity whose variations make up matter waves is called the wave function (ψ).
2. What is meant by normalized wave function?
∞ ∞
If |𝜓|2 =1 then it must be true that ∫−∞|𝜓|2 𝑑V = ∫−∞|𝜓|2 𝑑𝑥𝑑𝑦𝑑𝑧 = 1 – Normalization.
3. What is a quantum state? What does a quantum state represent? What happens to a quantum state
when it is measured?
A quantum state is the complete mathematical description of a physical system. It is a vector in
a Hilbert space. When a measurement is performed, the quantum state collapses as described by
the Born rule.
4. What does the Hamiltonian operator represent in quantum mechanics?
2
̂ = −ℏ 𝛻 2 + V is called Hamiltonian operator
H 2𝑚
𝜕
5. What does applying 𝑖ℏ 𝜕𝑡 to a wave function yield?
̂ = −ℏ 𝜕 or 𝑖ℏ 𝜕 is called energy operator – eigen energy
E 𝑖 𝜕𝑡 𝜕𝑡
6. What is a Hilbert space in quantum mechanics?
𝓗 is a Hilbert space, it is a complete complex vector space with an inner product.
7. What is Dirac notation used for?
It is used to describe quantum states as "kets" (|ψ⟩), their complex conjugates as "bras" (⟨ψ|), and their
inner products, which yield probabilities, as "bra-kets" (⟨ψ|ψ⟩).
8. What is quantum tunneling?
𝐓 = 𝐞−𝟐𝛋𝟐 𝐋
Even when E < V0, there is always a certain probability that particle may cross the barrier and go
over to the other side. This effect is called barrier penetration or “tunnelling effect”.
Transmission probability, T will decrease with (i) increase of width L of the barrier and (ii) decrease
of E, the kinetic energy of the particles.
9. What does the de Broglie hypothesis state?
All moving particles behaves as waves
ℎ ℎ
de-Broglie wavelength, 𝜆 = 𝑝 = m𝑣
10. What are the two quantities involved in Heisenberg’s uncertainty principle?
It is impossible to know both, the exact position and the exact momentum of an object at the same
time”.
ℏ
∆𝒙∆𝒑𝒙 ≥ 𝟐
, Essay-type questions
1. (i) Apply Schrodinger’s equation for a quantum mechanical particle confined in a potential box
defined as V(x) = 0 for 0 ≤ x ≤ a and V(x) = ∞ and obtain the energy Eigen values and Eigen functions
for this particle in the ground, 1st and 2nd excited states.
Potential energy V(x) of the particle is
V(x) = 0 0<x<L
V(x) = ∞ x = 0, L
Particle cannot have an infinite amount of energy; it cannot exist outside the box.
𝜓(x) = 0 x ≤ 0 and x ≥ L
𝜓(x) ≠ 0 0<x<L
Apply Schrӧdinger’s time independent wave equation within the box,
𝒅𝟐 𝝍(𝒙) 𝟐𝒎𝑬
+ ℏ𝟐 𝝍(𝒙) = 𝟎 →(1)
𝒅𝒙𝟐
Equation (1) has the possible solution of
𝜓(x) = Asin 𝜅 𝑥 + B cos 𝜅 𝑥 →(2)
√2m𝐸
Where 𝜅 = ℏ , A and B are constants.
This solution is subject to the boundary conditions that 𝜓(x) = 0 for x = 0 and for x = L.
Since cos 0 = 1, the second term cannot describe the particle because it does not vanish at x = 0.
(B = 0).
Since sin 0 = 0, the sine term always yields 𝜓(0) = 0 at x = 0 as required, but 𝜓(x) will be 0 at x
√𝟐𝐦𝐄
= L only when 𝜿𝐋 = ℏ 𝐋 = 𝐧𝝅 where n = 1, 2, 3,…… →(3)
Energy of the particle can have only certain values called as Eigen values.
These Eigen values, constituting the energy levels of the system.
𝐧𝟐 𝝅 𝟐 ℏ 𝟐
From equation (3), 𝐄𝐧 = 𝟐𝐦𝐋𝟐 where n = 1, 2, 3,…… →(4)
For each allowable value of n the particle energy is different and thus energy is quantized.
Wave function 𝝍(x):
√2mE
𝜓n (𝑥) = A sin ℏ n 𝑥 →(5)
Substituting equation (4) in (5) for En gives
𝐧𝝅
𝝍𝒏 (𝒙) = 𝑨 𝒔𝒊𝒏 𝐋 𝒙 →(6)
Equation (6) gives Eigen functions corresponding to the energy Eigen values En.
The integral of |𝜓n |2 over all space is finite.
∞ L
∫−∞|𝜓n |2 d𝑥 = ∫0 |𝜓n |2 d 𝑥
∞ L
n𝜋𝑥
∫ |𝜓n |2 d𝑥 = A2 ∫ sin2 ( ) d𝑥
L
−∞ 0
Short Answer Type Questions
1. Define wave function Ψ
The quantity whose variations make up matter waves is called the wave function (ψ).
2. What is meant by normalized wave function?
∞ ∞
If |𝜓|2 =1 then it must be true that ∫−∞|𝜓|2 𝑑V = ∫−∞|𝜓|2 𝑑𝑥𝑑𝑦𝑑𝑧 = 1 – Normalization.
3. What is a quantum state? What does a quantum state represent? What happens to a quantum state
when it is measured?
A quantum state is the complete mathematical description of a physical system. It is a vector in
a Hilbert space. When a measurement is performed, the quantum state collapses as described by
the Born rule.
4. What does the Hamiltonian operator represent in quantum mechanics?
2
̂ = −ℏ 𝛻 2 + V is called Hamiltonian operator
H 2𝑚
𝜕
5. What does applying 𝑖ℏ 𝜕𝑡 to a wave function yield?
̂ = −ℏ 𝜕 or 𝑖ℏ 𝜕 is called energy operator – eigen energy
E 𝑖 𝜕𝑡 𝜕𝑡
6. What is a Hilbert space in quantum mechanics?
𝓗 is a Hilbert space, it is a complete complex vector space with an inner product.
7. What is Dirac notation used for?
It is used to describe quantum states as "kets" (|ψ⟩), their complex conjugates as "bras" (⟨ψ|), and their
inner products, which yield probabilities, as "bra-kets" (⟨ψ|ψ⟩).
8. What is quantum tunneling?
𝐓 = 𝐞−𝟐𝛋𝟐 𝐋
Even when E < V0, there is always a certain probability that particle may cross the barrier and go
over to the other side. This effect is called barrier penetration or “tunnelling effect”.
Transmission probability, T will decrease with (i) increase of width L of the barrier and (ii) decrease
of E, the kinetic energy of the particles.
9. What does the de Broglie hypothesis state?
All moving particles behaves as waves
ℎ ℎ
de-Broglie wavelength, 𝜆 = 𝑝 = m𝑣
10. What are the two quantities involved in Heisenberg’s uncertainty principle?
It is impossible to know both, the exact position and the exact momentum of an object at the same
time”.
ℏ
∆𝒙∆𝒑𝒙 ≥ 𝟐
, Essay-type questions
1. (i) Apply Schrodinger’s equation for a quantum mechanical particle confined in a potential box
defined as V(x) = 0 for 0 ≤ x ≤ a and V(x) = ∞ and obtain the energy Eigen values and Eigen functions
for this particle in the ground, 1st and 2nd excited states.
Potential energy V(x) of the particle is
V(x) = 0 0<x<L
V(x) = ∞ x = 0, L
Particle cannot have an infinite amount of energy; it cannot exist outside the box.
𝜓(x) = 0 x ≤ 0 and x ≥ L
𝜓(x) ≠ 0 0<x<L
Apply Schrӧdinger’s time independent wave equation within the box,
𝒅𝟐 𝝍(𝒙) 𝟐𝒎𝑬
+ ℏ𝟐 𝝍(𝒙) = 𝟎 →(1)
𝒅𝒙𝟐
Equation (1) has the possible solution of
𝜓(x) = Asin 𝜅 𝑥 + B cos 𝜅 𝑥 →(2)
√2m𝐸
Where 𝜅 = ℏ , A and B are constants.
This solution is subject to the boundary conditions that 𝜓(x) = 0 for x = 0 and for x = L.
Since cos 0 = 1, the second term cannot describe the particle because it does not vanish at x = 0.
(B = 0).
Since sin 0 = 0, the sine term always yields 𝜓(0) = 0 at x = 0 as required, but 𝜓(x) will be 0 at x
√𝟐𝐦𝐄
= L only when 𝜿𝐋 = ℏ 𝐋 = 𝐧𝝅 where n = 1, 2, 3,…… →(3)
Energy of the particle can have only certain values called as Eigen values.
These Eigen values, constituting the energy levels of the system.
𝐧𝟐 𝝅 𝟐 ℏ 𝟐
From equation (3), 𝐄𝐧 = 𝟐𝐦𝐋𝟐 where n = 1, 2, 3,…… →(4)
For each allowable value of n the particle energy is different and thus energy is quantized.
Wave function 𝝍(x):
√2mE
𝜓n (𝑥) = A sin ℏ n 𝑥 →(5)
Substituting equation (4) in (5) for En gives
𝐧𝝅
𝝍𝒏 (𝒙) = 𝑨 𝒔𝒊𝒏 𝐋 𝒙 →(6)
Equation (6) gives Eigen functions corresponding to the energy Eigen values En.
The integral of |𝜓n |2 over all space is finite.
∞ L
∫−∞|𝜓n |2 d𝑥 = ∫0 |𝜓n |2 d 𝑥
∞ L
n𝜋𝑥
∫ |𝜓n |2 d𝑥 = A2 ∫ sin2 ( ) d𝑥
L
−∞ 0
