Rotational Energy
For a molecule of $O_2$ a) Determine the lowest energy of rotation. b) What is the energy of the photon that will excite the molecule into its first excite state (starting… Click here to read more
For a molecule of $O_2$ a) Determine the lowest energy of rotation. b) What is the energy of the photon that will excite the molecule into its first excite state (starting… Click here to read more
Consider an operator defined as a product of bra and ket vectors:$\hat H =E (|\varphi_1><\varphi_1|-|\varphi_2><\varphi_2|-i|\varphi_1><\varphi_2|+i|\varphi_2><\varphi_1|)$ where $E$ is a real constant, and $\phi_i>$ form an orthonormal basis in the two… Click here to read more
Consider a system whose Hamiltonian is given by the matrix$H=ϵ\begin{pmatrix}0&-i&0\\i&0&2i\\0&-2i&0\end{pmatrix}$ a) If we measure energy, which values we will obtain?b) If the system starts in the state $(1/√3) \begin{pmatrix}1\\1\\1\end{pmatrix}$ what… Click here to read more
Please find the Gain of the Common Emitter amplifier. Use the given $h$ parameters. Assume $R_L=1 MOmega$. For a good approximation you just need to know $β=h_{fe}=100$ $V_b=20 V*10K/(10K+40K)=4V$First approx… Click here to read more
3. Prove the following identitya) the comutator of two Hermitian operators is anti-Hermitian. b) product of comutators $[AB, CD]=-AC\{D,B\}+A\{C.B\}D+-C\{D,A\}B+\{C,A\}DB$ a)$A$ is hermitic if $A=A^\dagger$ ; $A$ is antihermitic if $A=-A^\dagger$… Click here to read more
2. Use the following definitions $S_x =(\hbar/2)\{|+><-|+|-><+|\}$ and $S_y =(\hbar/2)\{-i|+><-|+i|-><+|\}$ to show that $S_z=S_x \pm iS_y$ Notation $|m><n|=|m,n|$$S_x=ℏ/2(|+,-| + |-,+|)$ and $S_y=-iℏ/2(|+,-| -|-,+|)$Therefore $S_+=ℏ*|+,-| =S_x+iS_y$ and $S_-=ℏ*|-,+|=S_x-iS_y$ 4. Use… Click here to read more
You are given two charges $q_0$ and $q$ situated on the z axis at $z=+a$ and $z=-a$. What are the monopole and dipole terms in the potential, when the distance… Click here to read more
Consider a matrix $A=\begin{pmatrix}0&0&1\\0&1&1\\-1&0&0\end{pmatrix}$ a) Find its eigenvalues and eigenvectors.b) Construct a system of three normalized orthogonal eigenvectors and verify the completness condition.c) Using eigenvectors of $A$ construct the transformation matrix… Click here to read more
Deduce the equation of the Isentropic (Adiabatic) transformation starting from the first principle. Assume an ideal gas. From first principle one has $d Q=d U+d W$ (Clausius sign convention:work… Click here to read more
For a square loop of side $a$ in a magnetic field $mathbf{B}=B_0[1-(x^2/l^2)]*hat x$ rotating with angular velocity $omega$ please find $E(t)$ induced in loop. In text it says that the… Click here to read more
Please show that the Magnetic vector potential of a solenoid is consistent with the required values of it’s divergence, curl and laplacian in all regions. $A_ϕ=μnI/2 s*ϕ ̂ $ inside… Click here to read more
Consider the ground state of Hydrogen.Treat the nucleus as infinitely heavy. Show that the fine structure constant $\alpha =e^2/(4\pi\epsilon_0\hbar c)$ determines the typical speed of the electron (relative to $c$)… Click here to read more
Consider a Hydrogen atom in a state described by the following wave function$\psi(r)=\frac{A}{\sqrt{\pi}}(1/a)^{3/2}e^{-r/a} +\frac{1}{2\pi}\frac{z-\sqrt{2}x}{r}*R_{21}(r)$ where $A$ is a real constant, $a$ is the Bohr radius, and R_{21}(r) is the radial… Click here to read more
A hydrogen atom is in a state described by the wave function $\Psi_{nlm}$ with $n=4$, $l=3$ and $m=3$. a) What is the magnitude of the orbital momentum of the electron in… Click here to read more
A particle of mass $m$ moves in the $x y$ plane in a potential described as$V(x,y)=\left\{\begin{matrix} m\omega^2y^2/2 & \text{for all y and } 0<x<a\\ \infty & \text{elsewhere}\end{matrix}\right.$ Using separation of… Click here to read more
Find the acceleration of a fixed Yo-Yo and a Yo-Yo in free fall (see the images). Moment of inertia is $I=MR^2/2$. What are the fall times and velocities for the… Click here to read more
For the given wavepacket$ψ(x,t)=A*e^{-(a^2 x^2+b^2 t^2+2abxt)}$ show that it satisfies the 1D wave equation and find its speed and direction of propagation. Wave equation is$(∂^2 ψ)/(∂x^2 )=1/v^2 (∂^2 ψ)/(∂t^2)$(Computations have… Click here to read more
For a system with 3 possible energies $0$, $ε$, and $2ε$ having the $\epsilon$ level double degenerate, please find the partition function, the average energy, its entropy and specific heat…. Click here to read more
Consider a system described by a Hamiltonian $H=p/^22+x^2/2$ which is written in the system of units such that $\hbar=1$, and $p=-i*(d/dx)$. At $t=0$ the system is in the state described… Click here to read more
1. We know that the $\mathbf{D}$ and $\mathbf{H}$ fields are defined as $\mathbf{D}=\epsilon_0 \mathbf{E}+\mathbf{P}$ and $\mathbf{H}=\mathbf{B}/\mu_0 -\mathbf{M}$, and the bound current density is given by $\mathbf{J_s}=\nabla \times \mathbf{M} +d\mathbf{P}/d t$. … Click here to read more