विभीय विश्लेषण का प्रयोग करके यह दिखाइए कि यह केप्लर के तृतीय नियम का पालन करते है।

The Hamiltonian for an electron at rest in a magnetic field Bo along the z-direction is , where . Given that the initial state of the system is  determine the state and  at al later time t.

i) Write down the angular momentum states |j,mj) for j = 2 and the eigenvalue of J² and J_₂ for each of these states.

ii) Determine J+ 2,1) and J_|2,1).

i) Show that the Hamiltonian for a simple harmonic oscillator can be written in terms of the raising and lowering operators as Ĥ = 

ii) Obtain the value of * (n|p^-2|n) for the eigenket | n) of t of the simple harmonic oscillator.

Derive the Heisenberg equations of motion for the simple harmonic oscillator.

A two state system has the orthonormal energy eigenkets IE₁) and IE2) with eigenvalues E_₁ and E₂. If the initial state of the system is given by, determine .

The orthonormal bases for a three-dimensional

Hilbert space is described by . The action of an operator Ô in this space is given by:

Obtain the matrix representation of this operator.

Show that a linear combination of the degenerate eigenvectors of an operator belonging to a particular eigenvalue of the operator is also an eigenvector belonging to the same eigenvalue.

Consider an electron in the state

where , are the Hydrogen atom eigenfunctions. Determine the normalization constant N and the expectation values of the energy, L² and L^z.

Calculate the probability that a simple harmonic oscillator in its ground state will be found beyond the classical turning points.

Consider a one dimensional potential well with an infinite barrier at x = 0 and a finite potential V for x ≥ L. Solve the Schrödinger equation for a particle of mass m inside the well with an energy E

Show that [x², px²] = 2ih(xpx +p _xx)

The wavefunction of particle of mass m confined to move in one dimension is ψ(x) = Nx exp(-ax), 0 ≤ x ≤∞. Calculate the normalization constant N and the expectation value of 1/x.

Calculate the probability current density for the wavefunction: w(x) = f(x)exp[ig(x)].

Estimate the kinetic energy of the neutrons in a neutron beam that can be used to probe lattice structures with an interatomic spacing of 0.3 nm. The mass of the neutron is 1.675 x 10-²⁷ kg.

Show that in the x - y plane, a rotation about the origin in an anti-clockwise

direction by an angle φ will move the points (x, y) to by the matrix 

Obtain all the proper subgroups of the permutation groups S₃ = {e,p1...,p5 }and their cosets.

Given: 

Construct the multiplication table for the group of permutations of {1,2,3}.

Determine the inverse Laplace transform of

A metal plate covering the first quadrant of the xy plane has its edge along y-axis insulated. The edge along x-axis is held at an initial temperature

Obtain the steady state temperature distribution as a function of x and y.