Quantum Mechanics 1

Explain the phenomenon of quantum tunneling and its application in quantum communication using x-ray beams from the Advanced Photon Source.

Consider a particle that is traveling towards a barrier with energy $E$ less than the potential barrier $U_0$. Classically, the particle would bounce off, but quantum mechanically, it can tunnel through. Calculate the tunneling probability for the particle using the boundary conditions and wavefunctions for the three regions separated by the potential barrier.

Solve the radial part of the Schrödinger equation for the hydrogen atom for different values of the angular momentum quantum number, $l$.

Using the raising and lowering operators (A+ and A-), determine the action of these operators on the eigenstates of the quantum harmonic oscillator.

Explain the properties and significance of raising and lowering operators in the quantum harmonic oscillator.

Solve the Schrödinger equation with the harmonic oscillator potential energy, $\frac{1}{2} kx^2$, to find the wave function $\psi(x)$. Consider potential solutions like $\sin(ax)$, polynomials (e.g., $ax^3$, $ax^4$), or exponential functions ($e^{ax}$).

Determine the allowed energies and wave function for a particle undergoing harmonic motion.

Using the Schrödinger equation, find the energies for a quantum harmonic oscillator, and relate them to the classical harmonic oscillator.

Solve the Schrödinger equation for the hydrogen atom by separating it into radial and angular parts. Given the radial equation: $\dfrac{1}{r} \dfrac{d}{dr} \left(r^2 \dfrac{dR}{dr}\right) + \left( E - \dfrac{Z^2 e^2}{4 \pi \epsilon_0 r} \right)R = 0$ and the angular equation: $\dfrac{1}{Y} \dfrac{1}{\sin \theta} \dfrac{d}{d \theta} \left( \sin \theta \dfrac{dY}{d \theta} \right) + \dfrac{1}{Y \sin^2 \theta} \dfrac{d^2 Y}{d \phi^2} = \alpha$, find the solutions for the functions $R(r)$ and $Y(\theta, \phi)$.

Solve the Schrödinger equation in one dimension using the method of separation of variables.

Derive the time-independent Schrödinger equation from the time-dependent Schrödinger equation by separating the wavefunction into spatial and temporal components, $\psi(x, t) = \phi(x) \cdot \xi(t)$, and demonstrate each step in the derivation.

Solve the angular part of the Schrödinger equation in three dimensions using spherical harmonics.

Solve the Dirichlet problem for Laplace's equation inside a sphere using separation of variables in spherical coordinates.

In this problem, we're dealing with a situation where the magnetic field is at some angle relative to the z-axis, but we've written the Hamiltonian in the spin up spin down basis. Part A: Obtain the matrix representation of $H$ in the spin up spin down basis. Part B: Find the eigenvalues and normalized eigenvectors of $H$. Part C: Verify that the normalized eigenvectors of $H$, denoted as $E_1$ and $E_2$, satisfy the Dirac completeness relation.

In this problem, we're dealing with a situation where the magnetic field is at some angle relative to the z-axis, but we've written the Hamiltonian in the spin up spin down basis. Part A: Obtain the matrix representation of $H$ in the spin up spin down basis. Part B: Find the eigenvalues and normalized eigenvectors of $H$. Part C: Verify that the normalized eigenvectors of $H$, denoted as $E_1$ and $E_2$, satisfy the Dirac completeness relation.

Determine the four quantum numbers (n, l, ml, ms) for the last valence electron of fluorine, which is the 2p^5 electron.

Find the four quantum numbers (n, l, ml, ms) for the last electron of iron, which is the 3d^6 electron.

Determine the four quantum numbers (n, l, ml, ms) for the fifth electron in the 4f subshell.

Discuss the idea of normalization and derive the constraint on the coefficients for a general spin-1/2 ket state, ensuring that it is properly normalized. The constraint is given by: if $\left| \psi \right\rangle = a \left| + \right\rangle + b \left| - \right\rangle$, then $1 = |a|^2 + |b|^2$.

Write the orbital diagram and electron configuration for nitrogen, which has an atomic number of 7.