Engineering question bank

Physics question bank
Unit 03 • Unit 04 • Unit 05 • Unit 06

Unit NO:- 03
1. Obtain the expression for energy when particle is confined to a rigid box.

To solve for the energy eigenvalues and wave functions of this system, we can apply the
time-independent Schrödinger equation:

schroedinger ′ s time independent wave equation

2m
∇2 ψ + . (E − V(P. E) )ψ = 0 − − − (1)
ħ2

P. E, V = 0 inside the box and particle move along x − direction
d2 ψ 2m
+ 2 . Eψ = 0 − − − (2)
dx 2 ħ
2mE
= K 2 − − − (3)
ħ2
d2 ψ
∴ 2 + K 2 ψ = 0 − − − (4)
dx
solution of equation 4 can be either a sine function or a cosine function.
hence general solution is ψx
= A sin sin (kx) + B cos cos (kx) − −(5) − −(A and B are constant )
ψ = 0 at x = 0 and ψ = 0 at x = L − − − (6)
putting ψ = 0 at x = 0 in equation (5)
0 = A.sin sin 0 + B.cos cos 0
B = 0 equation 5 witten as ψ = A sin sin kx − − − −(7)
putting second boundary condition i. e ψ = 0 at x = L in equation 7
0 = A.sin sin kL − −A ≠ 0 since then ψ = 0 at x = L
∴sin sin KL = 0 − − − (KL = nπ where n = 1,2,3, … )
nπ
∴k= − − − (8)
L
putting k from equation 3

2mE nπ
k= √ =
ħ2 L
2mE n2 π2 n2 π2 ħ2
= 2 or En = − − − −(9)
ħ2 L 2mL2

pg. 1

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, n2 h2 h
or En = 2
(as ħ2 = ) − − − (10)
8mL 2π



2. State Debroglie Hypothesis. Derive the Debroglie wavelength of electron.

The de Broglie wavelength of an electron can be calculated using the de Broglie wavelength
equation:

ℎ ℎ
𝜆 = =
𝑝 𝑚𝑜 𝑣


Where λ is the de Broglie wavelength, h is the Planck's constant (𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 6.626 ×
10−34 𝐽 · 𝑠), and p is the momentum of the electron.

To determine the momentum of an electron, we can use the equation:

𝑝 = 𝑚 × 𝑣

Where p is the momentum, m is the mass of the electron (𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 9.10938356 ×
10−31 𝑘𝑔), and v is the velocity of the electron.

However, for non-relativistic electrons (those with speeds much lower than the speed of light),
we can approximate the momentum as:

𝑝 ≈ 𝑚 × 𝑣

Substituting this approximation into the de Broglie wavelength equation, we get:

ℎ 12.26
𝜆= = 𝐴°
√2𝑚𝑜 𝑒𝑉 √𝑣

So, the de Broglie wavelength of an electron is inversely proportional to its velocity.

It's important to note that the de Broglie wavelength of an electron is significant when
considering its wave-like behavior, such as interference and diffraction. For electrons with high
velocities (relativistic speeds), additional factors need to be considered in the calculation of
their de Broglie wavelength.

3. Define Phase velocity and group velocity. Show that 𝐕𝐩 𝐕𝐠 = 𝐂 𝟐 .

pg. 2

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, 4. Derive Schrodinger’s Time independent wave equation.

Schrodinger's time-independent wave equation is a fundamental equation in quantum
mechanics that describes the behavior of a quantum system with a time-independent potential
energy. It is known as the time-independent Schrodinger equation or the stationary
Schrodinger equation.

The time-independent Schrodinger equation is given by:

h h
λ= =
p mv

p = mv = momentum of particle

d2 y 2
d2 y
=v . 2
dt 2 dx

∂2 ψ 2
∂2 ψ ∂2 ψ ∂2 ψ
= U [ 2 + 2 + 2]
∂t 2 ∂x ∂y ∂z

∂2 ψ
= u2 . ∇2 ψ
∂t 2

2
∂2 ∂2 ∂2
∇ = + + is called as a laplacian operator.
∂x 2 ∂y 2 ∂z 2

ψ(x, y, z, t) = ψ0 (x, y, z). e−iωt

ψ(r, t) = ψ0 (r). e−iωt

∂ψ
= −iωψ0 . e−iωt
∂t
∂2 ψ
= i2 ω2 ψ0 . e−iωt
∂t 2
∂2 ψ
or = ω2 ψ − − − −(i2 = −1 and ψ0 . e−iωt = ψ
∂t
∇2 u2 ψ = − ω2 ψ
2
ω2
∇ ψ+ 2 ψ=0
u
ω = 2πʋ and u = ʋ. λ − − − −(ʋ = frequency of wave)

pg. 3

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, ω 2πʋ 2π
= =
u ʋλ λ
2
4π
∇2 ψ + 2 ψ = 0
λ
Total energy (E) = kinetic energy + potential energy
1
E = mv 2 + V(P. E)
2
1 2 2
E= m v + V(P. E)
2m
p2
E= + V(P. E)
2m
p2 = 2m(E − V(P. E) )
4π2
∇2 ψ + 2 . 2m(E − V(P. E) )ψ = 0
h
h
putting =ħ
2π
2m
∇2 ψ + 2 . (E − V(P. E) )ψ = 0
ħ

In this equation:

- ħ is the Hamiltonian operator, which represents the total energy of the system and is given
by the sum of the kinetic energy operator and the potential energy operator.

- Ψ is the wave function of the system, representing the quantum state of the particle or
system.

- E is the energy eigenvalue associated with the particular state described by the wave
function.

5. State Heisenberg’s Uncertainty Principle. Calculate the Debroglie wavelength of 10
Kev electron.

Heisenberg's uncertainty principle states that it is impossible to simultaneously determine the
exact position and momentum of a particle with absolute certainty. The principle can be
mathematically expressed as:

h
Δx × Δp ≥
4π

Where:

Δx is the uncertainty in position,
pg. 4

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