Quantum Physics

Advance Physics

Quantuam Mechanics, Superconductivity, Electromagnetism

This page explains concepts at college level in the field of quantum mechanics, theory of relativity, superconductivty, Lagrangian method of equation of motion, Solid State Physics, Wave Particle Duality, Diffraction.

Definitions and Formula

• Photons: These are particle-like quanta of electromagnetic radiation which travel at the speed of light c with momentum p and relativistic energy E where p = h/λ and E = hc/λ. Here, h = Plank's constant and λ = wavelength of eletromagnetic radiation.
• de Broglie wavelength: Particle of matter with momentum p could act as a wave with wavelength λ = h/p. the de Broglie wavelength of a particle of matter is very small (as compared to its size) and difficult to measure.
• When the particle is treated as non-relativistic: E = mc² + E_K where E_K is the kinetic energy of non-relativistic particle given by E_K = p²/2m. Note p = mv and E_K = ½mv² = ½ m×p²/m² = p²/2m

Classical vs. Quantuam Physics

Superconductivity

• The resistivity of a superconductor is practically zero but only below certain temperature known as Critical Temperature, T_C.
• When superconducting materials are cooled below their critical temperatures in the presence of a magnetic field, the magnetic flux is expelled from the interior of the superconductor.
• Similarly, superconducting materials lose their superconducting behavior above a certain critical magnetic field, B_C which itself is temperature-dependent that is B_C = B_C(T).
  
  
• The value of the critical field limits the maximum [critical] current I_C that can be sustained in superconducting materials.
• Critical current in a superconducting wire of radius 'a' is given by I_C = 2πB_Ca/μ₀. This is the maximum current the wire can carry at any given temperature and still maintain superconducting features.

Length Contraction

A rod of length L is inclined at angle θ with x-axis [in the moving reference frame] at velocity 0.6c where c is the speed of light in vacuum. Calculate the angle observed in a stationary reference frame.


tanθ = L_Y / L_X.
Since rod the moving along X-axis, the length perpendicular to the direction of motion will remain constant as measured my stationary onserver. Thus: L_Y' = L_Y. The contraction in horizontal (X-) component of the length of the rod is given by


The red line shows rod as seen in moving reference frame and the blue line as observed in stationary reference frame. Thus, angle of inclination in stationary reference frame is given by:
  tanθ' = L_Y / L_X' = 1.25 * L_Y / L_X = 1.25tanθ

Superconductivity

A metal behaves as superconductive at 10 [K] and the critical magnetic field at 0 [K] is B₀ [A/m]. Calculate the magnetic field at 20 [K].

The critical temperature is T_C = 10 [K]. The relative between magnetic field and temperature is given by expression:

Proper - Improper Time Interval

The ground monitoring team of a spaceship moving at constant speed noted the time difference as per his stop-watch as 9 [s] when the location is shown to be 1.44E10⁹ [m]. Calculate the proper time interval

Note that the definition of proper time interval is "time interval measured in a reference frame where two events occcur at same place". Hence, a person monitoring time in space-ship will measure time at same place and will be the proper time interval, Δt. Thus, the time interval measured by the ground staff is improper time interval, Δt'. We have



v = 1.44x10⁹ [m] / 9 [s] = 1.8x10⁸ [m/s], Δt' = 9 [s].
Δt = Δt' * 0.8 = 7.2 [s].
Note that a moving clock runs slower than a stationary clock and hence "proper time interval" ≤ "improper time interval".

Relativity and Impulse-Momentum Theorem

Calculate the relativistic speed of a particle of rest mass m₀ after time t when a constant force F acts on it. The particle stats with velocity v₀ at t = 0.

From Newtons's law: dp/dt = F. Here, the momentum p = m(v) * v where m(v) is mass of the particle at speed v. Thus:

Particle in Box

Calculate the number of bound states for energy (potential) level E = 11h²/[8mL²] where L is the length of the box and h is the Planck's constant.

The potential well for a particle in a box is shown below


The energy level at quantum state 'n' is given by


Thus, the equation E_n < E = 11h²/[8mL²] is satisfied for n = 1, 2 and 3. Hence, number of bound states is 3.

Note that the energy in ground state E₁ > 0, whereas the classical physics assigns a value zero to minimum energy level. This excess energy of the ground state (with respect to the classical minimum) is known as zero point energy. Thus, the kinetic energy hence the momentum of a bound particle cannot be reduced to zero. The minimum value of momentum is E₁ = p²/2m.

Heisenberg's uncertainty principle

A neutron is moving at velocity 9*10⁷ [m/s]. What is the minimum uncertainty in position of this particle.

The maximum uncertainty in position is size of the box, that is Δx = L. Thus, Δx.Δp ≈ h. Similary, minimum uncertainty in position will correspond to maximum uncertainty in momentum.

Momentum of the neutron particle is given by
  p = m_n * v = 1.674929 x 10^-27 [kg] * 9 x 10⁷ [m/s]
  p = 5.165 x 10^-19 [kg-m/s] = maximum uncertainty in momentum.

Thus: minimum uncertainty in position, Δx_MIN = h / Δp_MAX
  Δx_MIN = 6.626 x 10^-34 [kg-m²/s] / 5.165 x 10^-19 [kg-m/s] = 1.283 x 10^-15 [m].

Probability Distribution: Particle in Box

A particle is located in a box of length L and is assumed to be in ground state. Determine the probability of finding the particle in the region 0.25L to 0.75 L.