Quantum Mechanics Answers

If I have two entangled particles, one of which falls into a black hole, do they still remain entangled? If so does this imply that I could build a device that I could use to monitor inside a black hole?

3) (a) Suppose, I have normalized the wave function at some point of time. The wave function evolves with time according to time dependent Schrodinger equation. How do

I know that the wave function remains normalized after some time?

[Hint: Show that (x, t)|²dx = 0]

(b) Show that

d(p)

dt

=

dx

i.e. expectation values follow Newton's law.

2) Consider a 2D infinite potential well with the potential U(x, y) = 0 for 0 ≤ x ≤ a & 0 ≤ y ≤ B, and U(x, y) = ∞, otherwise. Solve the time-independent Schrodinger equation, and find the normalized wavefunction and the corresponding energies.

Tritium 3H (mt = 3.01605u) has a half-life of 12.3 years and releases 0.0186 MeV of energy per decay. What is the reason energy is released for a 4.1 g sample of Tritium?

In the amusement park ride Mr. Freeze at Six Flags, riders are uniformly accelerated from rest by magnetic induction motors along a 70 meter horizontal track in just 5 seconds. While accelerating, friction and air drag exert 500N of force on the train. The train then coasts through the loops and turns for the remainder of the ride. A typical train loaded with passengers has a mass of 2500 kg.

The wave function Ψ(x, t) = A exp [i(k1x − ω1t)] + A exp [i(k2x − ω2t)] is a superposition of two free-

particle wave functions. Both k1 and k2 are positive.

(a) Show that this wave function satisfies the Schodinger equation for a free particle of mass m.

(b) Find and plot the probability distribution function for Ψ(x, 0).

I have just recently been introduced to the Kalmeyer-Laughling wavefunction

$   \psi(s_1,\dots,s_N) \propto \delta_s \prod_{i<j} (z_i-z_j)^{\frac{s_is_j}{2}-\frac{1}{2}}\prod_k e^{\frac{i\pi}{2}(k-1)(s_k+1)}$

where we consider a lattice in two dimensions, and on each lattice site there is a spin-1/2 particles in either state up or state down, $s_i = \pm 1$. $\delta_s$ is 1 if the sum of the spin is zero, zero otherwise. I am bit confused about the $k$, but i believe it is simply a product over every lattice site?

I want to rewrite the wavefunction using occupation numbers instead, so $s_j = 2n_j-1$ where $n_j$ is the number of particles on site $j$. I should be able to get the following form:

$  \psi(n_1,\dots,n_N) \propto \delta_n \prod_{i<j}(z_i-z_j)^{2n_in_j}\prod_{i\neq j}(z_i-z_j)^{\alpha n_i}$

I tried rewriting the exponential function in the original form:

$\prod_ke^{\frac{i\pi}{2}(k-1)(s_k+1)} = \prod_ke^{\frac{i\pi}{2}(k-1)(2n_k)} = \prod_ke^{i\pi(k-1)n_k} $

 (a) Write down the Schrodinger equation for the electron in a hydrogen atom.

(b) Considering the central nature of the potential of the electron in hydrogen atom show that the solution of the Schrodinger wave equation is of the form:

Ψ(𝑟, 𝜃,𝜙) = 𝑅(𝑟)Θ(𝜃)Φ(𝜙), where Φ(𝜙)Θ(𝜃)𝑅(𝑟) satisfies the following differential equations

1. The components of the angular momentum operator in terms of the components of position operator and the linear momentum operator are given as:

𝐿̂ 𝑥 = 𝑦̂𝑝̂𝑧 − 𝑧̂𝑝̂𝑦 , 𝐿̂ 𝑦 = 𝑧̂𝑝̂𝑥 − 𝑥̂𝑝̂𝑧 𝑎𝑛𝑑 𝐿̂ 𝑧 = 𝑥̂𝑝̂𝑦 − 𝑦̂𝑝̂𝑥 ,

where the symbols have their usual meanings.

(a) Show that [𝐿̂ 𝑥, 𝐿̂ 𝑦] = 𝑖ℏ𝐿̂ 𝑧 𝑎𝑛𝑑 [𝐿̂2 , 𝐿̂ 𝑧 ] = 0. State other similar commutation relations.

(b) Define angular momentum ladder operators 𝐿̂±

(i) Show that [𝐿̂2 , 𝐿̂±] = 0 and [𝐿̂ 𝑧 , 𝐿̂±] = ±ℏ𝐿̂±.

(ii) Hence, show that |𝑙 𝜄𝑚𝜄 ⟩ = 𝐿̂±|𝑙, 𝑚⟩ is eigenstates of 𝐿̂2 and 𝐿̂ 𝑧 . What are the values of 𝑙 𝜄 and 𝑚𝜄 ?

Magnetic field required to remove retentivity of a magnetic material is called as