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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.


Superconductivity

  • The resistivity of a superconductor is practically zero but only below certain temperature known as Critical Temperature, TC.
  • 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, BC which itself is temperature-dependent that is BC = BC(T).
    Critical Magentic Field Superconductors
  • The value of the critical field limits the maximum [critical] current IC that can be sustained in superconducting materials.
  • Critical current in a superconducting wire of radius 'a' is given by IC = 2πBCa/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.
Length contraction of moving rod

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

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' = LY / LX' = 1.25 * LY / LX = 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 TC = 10 [K]. The relative between magnetic field and temperature is given by expression:
Critical Magentic Field Superconductors
Thus:   BC(20 K) = B0 * (1 - 1/4) = 0.75B0. Note that the magnetic field is specified in terms of [A/m] instead of standard unit of [Tesla]. Here, the specified value is actuall B/μ0 where μ0 is the permeability of vacuum having units of [T-m/A].


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

Proper - Improper Time Interval

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:

Relativistic Speed and Impulse-Momentum Therorem


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

Potetial Well - Particle in a box
The energy level at quantum state 'n' is given by
Potetial Well - Energy Quantum States

Thus, the equation En < 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 = mn * 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, ΔxMIN = h / ΔpMAX
  ΔxMIN = 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.

The probability distribution function (also known as wave function) is given by

Probability Distribution - Particle in a box
The ground state, 'n' = 1 and the square of the wavefunction is related to the probability of finding the particle in a specific position for a given energy level. Thus, cumulative probability between 0.25 to 0.75L is given by
Cumulative Probability - Particle in a box

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