1
vote
2answers
74 views

Given eigenvalues of $\vec l^2$ and $\vec s^2$, calculate the eigenvalue for $\vec j^2$

There was an exam question that read approximatly: Let $\vec j = \vec l + \vec s$. Given eigenvalues of $\vec l^2$ and $\vec s^2$, calculate the eigenvalue for $\vec j^2$. We came up with $$\vec ...
0
votes
0answers
34 views

Writing Schrodinger equation with central potential in Atomic unit

I'm struggling to write Schrodinger equation with a central potential in Atomic unit. $$ ...
0
votes
3answers
55 views

Superposition principle

If $S=(v_{1},v_{2}......v_{n})$ is a basis for vector Space V, then every vector v in V can be expressed in the form of $v=c_{1}v_{1}+.......c_{n}v_{n}$ in an unique way. Explain the significance of ...
0
votes
1answer
46 views

Modern physics photoelectric effect [closed]

In a photo electric experiment, energy of photon is 5eV incident on a metal surface. They liberate electrons which are just stopped by an electrode at a potential of -3.5V w.r.t the metal. The work ...
0
votes
0answers
33 views

Uncertainty in the hydrogen ground state [closed]

The following question is put forward by one of my students. 1.In spherical polar coordinates, can radial momentum have the form $(h/2\pi i)[\partial/\partial r - 1/r]$? 2. If it is so, what is the ...
0
votes
0answers
36 views

Finite Square Well Inside an Infinite Square Well

Ok here's a potential I invented and am trying to solve: $$ V(x) = \begin{cases} -V_0&0<x<b \\ 0&b<x<a \\ \infty&x>a \\ \end{cases}$$ and $V(-x) = V(x)$ (Even ...
1
vote
2answers
69 views

Idempotent Operators

If $P$ is an idempotent operator, $P^2 = P$ and we have a vector $|\psi\rangle$, $P\neq1$, and the relation $$PL |\psi\rangle = L|\psi\rangle,$$ what conclusions can we draw about $L$, which is a ...
1
vote
1answer
79 views

I am trying to calculate how $<r>$ in the hydrogen atom evolves with time

I am working on the Hydrogen atom and I was trying to calculate $\frac{d<r>}{dt}$ using $$\frac{d<r>}{dt} = \frac{i}{\hbar} <[\hat{H} , \hat{r}]>.$$ Here $r = \sqrt(x^2 + y^2 + z^2)$ ...
3
votes
1answer
70 views

Poles for a particle scattered in a delta potential

I am working on problem a professor gave me to get an idea for the research he does, and have hit a point where I'm having a difficult time seeing where I need to go from where I'm at. I would also ...
1
vote
2answers
40 views

Angular momentum for 3D harmonic oscillator in two different bases

I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\rangle$$ or, if solved in the spherical coordinate system: ...
1
vote
0answers
41 views

CPT symmetries for a free Klein-Gordon equation and in minimal coupling

I'm studying for an exam on relativistic quantum mechanics and one of the issues to prepare is about symmetries of Klein-Gordon equation concerning $C$, $P$, $T$ transformations for a free field, and ...
2
votes
1answer
91 views

Divergent solution in time-dependent Schrödinger equation

if I transform the time-dependent Schrödinger equation without a potential I get: $$ - \hbar \omega \psi(\omega,x) = \frac{- \hbar^2}{2m} \frac{\partial^2 \psi(\omega,x)}{\partial x^2}$$ The ...
2
votes
0answers
32 views

WKB approximation in two dimensions

Does anybody know how to implement the WKB approximation for the two-dimensional Schrodinger equation with a harmonic oscillator potential: $\frac{1}{2}\Biggl[-\biggl(\frac{\partial^2}{\partial ...
1
vote
1answer
24 views

Expectation value and Dispersion of an Operator

Suppose we have an operator $Q$ with eigenvalue $q$. Expectation value is $\langle Q \rangle$ and dispersion $D(Q) = \sqrt{\langle \left( Q - \langle Q \rangle \right)^2 \rangle} $. I want to find ...
3
votes
1answer
34 views

Probability of measuring momentum [closed]

Suppose we have this wavefunction: $$ \psi = A \left( cos(kx) + cos (2kx) \right) $$ I have to find the possible results of measurement of momentum and their probabilities. Attempt For a momentum ...
1
vote
1answer
33 views

Help needed to interpret question - Spin States of electron pair in Helium?

For the last part, I'm not sure what they mean by "explain how to form eigenstates of the total spin $\hat S^2$ and $S^z = S_1^z + S_2^z$. Are they simply referring to the spin singlet and tripplet ...
2
votes
0answers
57 views

Proof involving the fine-structure Hamiltonian of the Hydrogen atom

Given the perturbed Hamiltonian of the Hydrogen atom: $$ ...
1
vote
0answers
21 views

Magnetic moment with external magnetic field on a lattice?

Consider a system in which atoms are located in a regular lattice, each atom having a spin $1/2$ and an associated intrinsic magnetic moment $\mu_0>0$. Assume that each atom interacts only weakly ...
0
votes
0answers
60 views

Calculation of energy eigenvalues of $\hat{x}^4$

I would appreciate help in calculating the energy eigenvalues associated with $\hat{x}^4$, with $\hat{x}$ expressed using the ladder operators for harmonic oscillators. $\hat{x} = ...
2
votes
0answers
40 views

Proving that Measurement increases von Neumann entropy

Let $V$ be a finite dimensional complex inner product space. Let $\mathcal{M}$ be the classical sample space of measurement outcomes that may occur in a given experiment, and $M_\mu$, $\mu \in ...
0
votes
0answers
21 views

Particle in a box under harmonic driving

Is the particle in a box under harmonic driving electric field solvable analytically? Here is the Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} ...
1
vote
1answer
79 views

Expectation value of an operator

Suppose we have: $$ \hat{Q}|\psi_1\rangle=q_1|\psi_1\rangle \\ \hat{Q}|\psi_2\rangle=q_2|\psi_2\rangle $$ with $q_1 \neq q_2$. Then consider the state: $$ ...
0
votes
0answers
44 views

Quick question on Perturbation theory - How do I evaluate this probability?

We know that hamiltonian for interaction between an electron and external field along $z$ is: $$\hat H = \frac{e\hbar B}{2m}\hat \sigma_z = \frac{\hbar \omega}{2} \hat \sigma_z $$ This has energy ...
3
votes
1answer
124 views

Calculating probability current for scattering problem

I'm trying to calculate the probability current for a scattering problem. The potential is $V = V_0 > 0$ in $x>0$, with $E>V_0$ So I have in the region $x \le 0$: $$\psi = \exp(ikx) + R ...
1
vote
0answers
12 views

Coarse-graining on a second channel decreases mutual information?

Let $X_1,B_1,X_2,B_2$ and $Y_1,A_1,Y_2,A_2$ and $C_1$ and $C_2$ be binary random variables. Suppose: $I(X_2:B_2|C_2=0)+I(Y_2:A_2|C_2=1) \leq 1$. This can be thought of as a bound on the capacity ...
2
votes
1answer
76 views

Does the average momentum vanish for an eigenstate of the simple harmonic oscillator?

Suppose we have a simple harmonic oscillator, let's consider the ground state, $|0\rangle$ and the first excited state $|1\rangle$. $\langle 0|\hat p|0 \rangle$ is zero right? Since the particle can ...
2
votes
1answer
58 views

Relation between electric potential and wavelength of an electron

"An electron that is accelerated from rest through an electric potential difference of $V$ has a de Broglie wavelength of $\lambda$. Investigate the relationship between $V$ and $\lambda$." I had two ...
-1
votes
1answer
48 views

Calculations with operators - Proof: Equation of Operators [duplicate]

I have a problem in Quantum mechanics 1 with Operators. I have to prove the following equation. I tried it for about 4 hours without any result: Condition: $[[\hat A,\hat B],\hat A]=[[\hat A,\hat ...
1
vote
1answer
39 views

Nuclear shell model - finite square well

I am trying to make a simplified approximation and solve Schrodinger equation in the finite square well to model the nucleus of Ca (shell nuclear model). The potential is $ V(r) = -V_0$ for ...
1
vote
1answer
50 views

Atom state vectors kets

An atom with two energy levels has 2 states (excited and ground), represented by kets $|e\rangle$ and $|g\rangle$ respectively. The atom has energy $\frac{1}{2}E_\theta$ when excited and ...
0
votes
0answers
7 views

Determing what is differnece beetwen eigenvalues for attractive Coulomb Field (Hydrogen) calculated by exact method and WKB aproach

I have task to compare eignevalues gained by exact calculation on electron in Coulomb attractive field (hydrogen) and gained by WKB method.. As far as I get, eigenvalues gained with exact method are ...
1
vote
0answers
38 views

Quantum oscillator, position mean value problem

A quantum harmonic oscillator of mass $m$ and frequency $\omega$ is at time $t=0$ in the state: $$ \left|\psi(t)\right> = \sum_{n=N-\Delta N}^{N+\Delta N}\left|n\right>\frac{1}{\sqrt{2\Delta N ...
1
vote
0answers
49 views

Spin 1/2 particles hamiltonian, addition of angular momentum confusion

Suppose I want to compute $S^{1}_z -S^{2}_z$ on a singlet state $|0,0>$. (where $S^{i}_z$ are two particles' spin operators). $$|0,0> = \frac{1}{\sqrt{2}} (|\frac{1}{2},-\frac{1}{2}> - ...
0
votes
0answers
77 views

1D Infinite Square Well: Box Suddenly Increases in Size. How treat this?

I am currently working my way through John S. Townsend book "A Fundamental Approach to Modern Physics" (ISBN: 978-1-891389-62-7). Exercise 3.12 (p.111) is about the 1D infinite square well. The box ...
0
votes
0answers
48 views

Center of mass coordinates in Lagrangians and Laplacians

Is there a quick nice and easy way to write Lagrangian's and the classical/quantum Laplacian operator in terms of center of mass coordinates? The algebra is so involved and it has me confused about ...
0
votes
0answers
66 views

What is the eigenvalue of $J_z$?

In the calculation of the Zeeman Effect, the most important calculation is $$\langle J_z + S_z\rangle.$$ Suppose we want to find the Zeeman Effect for $(2p)^2$, meaning $l = 1$. In Sakurai's book, ...
1
vote
1answer
60 views

Manipulation with braket notation

I am still getting used to the braket notation. Is this manipulation correct? $$ \frac{-\hbar^2}{2m}\int_{-\infty}^{\infty}\frac{\partial^2\phi_n^*}{\partial z^2} \phi\,dz = ...
1
vote
0answers
69 views

Particle In a Box and momentum, velocity

So on a homework assignment, we are give the width of a well, $a$, and the mass of the particle $m$ and we want to find the average velocity of the particle at the n=1 state. So here is my attempt at ...
2
votes
1answer
91 views

Particle in a 1D Box with Symmetric potential: How find solutions?

I am working on a problem in which I shall find the normalised solution to the 1D particle in a box. Solving for the particle in an asymmetric potential is quite straight forward, but I run into ...
1
vote
2answers
94 views

How to determine a strong or weak interaction (Strange particles)?

As you know $\pi^-$ meson + proton ---> $K^+$ meson + $\Sigma^-$ particle. (AntiUp,down) + (up up down) --> (up antistrange) + (down down strange) I know that the quark number has be conserved in ...
1
vote
0answers
60 views

How to make a base tranformation for a linear operator in QM? [closed]

I have 2 bases A and B with the following kets: Base A: $|a_1\rangle$ and $|a_2\rangle$ Base B: $|b_1\rangle = \frac{1}{\sqrt2} \cdot(|a_1\rangle + i\cdot|a_2\rangle)$ $|b_2\rangle = ...
0
votes
3answers
94 views

Ground state of hydrogen atom

My interpretation: When we have no angular momentum, the potential well looks like this, my question is: How do you find the point where the wavefunction penetrates its classical forbidden region, ...
4
votes
2answers
115 views

Are nodes and orbitals in atoms simply probability distribution clouds or are they of any physical relevance?

I fail to understand what the electron clouds actually signify. Such as the $p$ orbitals, which have a dumbbell like shape. Now I am aware that they aren't actual trajectories of electrons, but what ...
0
votes
1answer
75 views

Quantum Mechanics - Finding momentum probability density [closed]

everyone. I got a bit stuck on 2(iii), this is supposed to be a easy question, but i don't know how you get the square term? I thought you just do the Fourier transform, but then I got some ...
1
vote
0answers
46 views

Quantum Rigid Rotor Perturbation

As the title says, I have a rigid rotor with a perturbation given below $$H=\frac{L^2}{2I}-\alpha B L_z.$$ So I know that the eigenvalues of $H$ will be $\ell(\ell+1)/2I -\alpha B m$ where $m$ is our ...
7
votes
2answers
336 views

Quantization of a particle on a spherical surface

Suppose we have a particle of mass $m$ confined to the surface of a sphere of radius $R$. The classical Lagrangian of the system is $$L = \frac{1}{2}mR^2 \dot{\theta}^2 + \frac{1}{2}m R^2 \sin^2 ...
4
votes
4answers
252 views

Evaluation of expectation values

I will denote operators with hats. Suppose we got an operator of the form $i[\hat p, \tan^{-1}(e^{\hat x})]$ and we want to calculate the amplitude for a transition from a state $|p_i\rangle$ to the ...
3
votes
0answers
73 views

How to quantize a rubber band stretched between two poles? [closed]

Consider a classical non-relativistic material string obeying Hooke's law stretched between two poles with either Neumann or Dirichlet or periodic boundary conditions and subject to either traversal ...
1
vote
0answers
54 views

Galilean Transform

I tried to solve a problem using two different ways and I had some trouble, the problem is: We define a symmetry transform of the expected value of $\vec{P}$ like this: $$\langle \psi|\vec{P}|\psi ...
-1
votes
1answer
82 views

How does $p_x$ commute with $p_y$, i.e. $[p_x,p_y]=0$? [closed]

I know it's a simple and basic question but would someone show me how to evaluate $[\hat{p}_x,\hat{p}_y]$?