Quantum Mechanics Answers

A particle of mass M, initially at rest decays into two particles with rest masses m1 and m2 respectively. Show that the total energy of the mass m1 is

E1= c^(2) [M^(2)+ m1^(2) - m2^(2)] / 2M

c= speed of light

Discuss some applications of Legendre polynomial in physics. Derive in detail
Spherical harmonics Laguerre polynomials

Discuss some applications of Legendre polynomial in physics. Derive in detail
Spherical harmonics Laguerre polynomials.

using the equipartition of energy concept estimate the ideal gas Cp(not Cv) in units KB for the low temperature regime where no vibrational modes are activated(if you calculate Cp=42KB, enter 42 as your answer )

Show that
i. σ’x2 = σ’y2 = σ’z(2) = 1
ii. [σ’x , αx ] = 0 ,
[σ’x , αy ] = 2i αz &
[σ’x , αz] = -2i αy

Q3. In Dirac’s theory, the probability current density is defined by the relation j(r, t) = CΨ* αΨ ,
where Ψ is the four component wave vector. Write the relations for jx, jy, jz in terms of the
component of Ψ i.e.
J(r, t) = Ψ*α Ψ ; jx = C Ψ* αx Ψ

A time t=0 the state vector |ψ› = 1/√2 [ |ϕ1› + |ϕ2›]
It is given as Hamiltonian is defined as H |ϕn› = n^2

Electrons travelling through matter may be diffracted. Describe and interpret the evidence provided by this observation about the nature of electron. Explain why a person off mass 70kg running at 5ms fails to show diffraction effects when passing through an open door of width 1.0m.

Q#5 An electron in a circular orbit about a proton can be described by classical mechanics if its angular momentum L is very much greater than h. Show that this condition is satisfied if the radius of the orbit r is very much greater than the Bohr radius a0 , i.e if

r>>a0 = 4πϵh2/e2me