Tagged Questions
2
votes
1answer
40 views
Time evolution of a quantum state
I have another point in QM that I would like clarified. Suppose $$\{|n\rangle\}$$ is a set of eigenstates of both the Hamiltonian $H$ and another operator $\hat O$ corresponding to an observable also. ...
-2
votes
0answers
27 views
Finding Clebsh-Gordan coefficient [closed]
Two electrons , their angular quantum number =1 . Find the inner product of < 2 0 | 00 > ?
-2
votes
0answers
29 views
The Hartree solution of two harmonic oscillator coupled by potential $V \propto ({\bf r}_1-{\bf r}_2)^2$ [closed]
$H={\bf p}_1^2+{\bf p}_2^2+{\bf r}_1^2+{\bf r}_2^2+x({\bf r}_1-{\bf r}_2)^2$.
$x$ is the coupling factor.
0
votes
1answer
59 views
Periodic boundary condition on a Wave Function of a Particle in a Box
Until now solving the Schrodinger Equation for a particle in a box was relatively easy because the boundaries conditions imposed zero value on the wave function at the boundaries. But now I must find ...
0
votes
0answers
76 views
Prove that the position operator is $\hat{x} = i\hbar \frac{d}{{dp}}$ in the momentum representation [closed]
Proof that: $x = i\hbar \frac{d}{{dp}}$
I did this, could you tell me if I am false or true
$\begin{array}{l}
x{e^{\frac{{ipx}}{\hbar }}} = - i\hbar \frac{{d{e^{\frac{{ipx}}{\hbar }}}}}{{dp}} = ...
1
vote
2answers
43 views
Time evolution of Gaussian wave packet
I'm slightly confused as to answer this question, someone please help:
Consider a free particle in one dimension, described by the initial wave function
$$\psi(x,0) = ...
0
votes
0answers
37 views
Question regarding operators and cylindrical coordinates
I have the following problem in my hand:
I need to arrive from the Cartesian expression $$x_{j}{\partial_{k}}x_{j}{\partial_{k}}-x_{j}{\partial_{k}}x_{k}{\partial_{j}}$$
to this expression:
...
1
vote
1answer
86 views
Matrix representation of state
This is a quantum mechanics question, I don't quite understand what it's getting at...
Suppose the we have a state described by $|1\,\,\, m\rangle$. Let its matrix representation be $\vec u$. ...
1
vote
2answers
71 views
Grover algorithm $R_D$ Circuit
I need sketch two circuits to understand Grover algorithm. The first is the operator $R_f$ and another is the operator $R_D = H^{\otimes n}(2|0\rangle\langle0|-I)H^{\otimes n}$. I get the first ...
0
votes
2answers
54 views
Electron in an infinite potential well
Does this problem have any sense?
Suppose an electron in an infinite well of length $0.5nm$. The state of the system is the superposition of the ground state and the first excited state. Find the ...
0
votes
0answers
54 views
The gauge-invariance of the probability current
It is simple to show that under the gauge transformation $$\begin{cases}\vec A\to\vec A+\nabla\chi\\
\phi\to\phi-\frac{\partial \chi}{\partial t}\\
\psi\to \psi ...
1
vote
1answer
46 views
Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$
I just finished deriving the commutators:
\begin{align}
[\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\
[\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\
\end{align}
On the ...
3
votes
2answers
185 views
Coherent State, Unitary Operators, Harmonic Oscillator
Consider the operator:
$$O = e^{\theta(a^\dagger b - b^\dagger a)}$$
where $\theta$ is a constant.
$O$ is a unitary operator.
$a$, $a^\dagger$, $b$, and $b^\dagger$ are ladder operators for two ...
1
vote
1answer
52 views
Eigenfunctions in a harmonic oscillator
This assignment is about the one dimensional harmonic oscillator (HO).
The hamiltonian is just as you know from the HO, same goes for the energies, but I get that the wavefunction of the particle, at ...
2
votes
1answer
61 views
Time evolution operator to find expectation value
I have a state $\Psi (x,0) = \sum_{n=0}^{\infty} c_{n}u_n(x)$ and want to find the expectation value of any observable A at time t, $\langle \Psi(t)|\hat{A}|\Psi(t)\rangle$.
I know that I should ...
4
votes
2answers
94 views
Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$
I know how to derive below equations found on wikipedia and have done it myselt too:
\begin{align}
\hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\
\hat{H} &= ...
2
votes
2answers
64 views
Hamiltonian of Harmonic Oscillator with Spin Term
We have the usual Hamiltonian for the 1D Harmonic Oscillator:
$\hat{H_{0}}=\frac{\hat{P^2}}{2m} + \frac{1}{2}m \omega \hat{X^2}$
Now a new term has been added to the Hamiltonian, $\hat{H} = ...
0
votes
0answers
33 views
Physical significance of effective wave function
In Yanhua Shih's book on quantum optics, the coherence functions are expressed in terms of effective wave function. Here are the expressions for single photon wave packets.
To derive the coherence ...
0
votes
1answer
41 views
Time Dependent HydroHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? gen Wave Function
1) How vwoulHow would I go about writing the time dependent wave function given the wavefunction at $t=0$?
go about writing the time dependent wave function given the wavefunction at $t=0$?
...
0
votes
1answer
59 views
Two qubits problem [closed]
Given the 2 qubit state:
(a/b) |00> + (c/b) |01> + (c/b) |10> + (d/b) |11>
What is the probability that 2 qubits are equal?
Thanks much!
0
votes
0answers
62 views
What does this notation mean in terms of quantic numbers, and how to imagine the electrons in this quantic system? (Helium $2^1$ $P$ and $2^3$ $P$)
Helium atom in the $2^1$ $P$ and $2^3$ $P$ excited states
Now I'm guessing that 1 electron should be considered in the 1s state, but what about the other?
Should I consider the other as simply ...
3
votes
1answer
138 views
Schrödinger equation for a harmonic oscillator
I have came across this equation for quantum harmonic oscillator
$$
W \psi = - \frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi
$$
which is often remodelled by defining a new ...
0
votes
1answer
32 views
Violation of the Normalization Constraint?
Say we have two qubits $|a\rangle$ and $|b\rangle$ both initialized to $|0\rangle$. We then apply the rotation gate $R_{x}(\frac{\pi}{2})$ of matrix representation
$\left( \begin{array}{}
...
3
votes
2answers
116 views
Quantum commutator
I'm given this commutator:
$$\left[PXP,P\right]$$
Being $P\psi=-i\hbar\partial_x\psi$, and $X\psi=x\psi$
I've solved it in two ways, the first one is just aplying the commutator to some function ...
1
vote
1answer
84 views
Finding the wavelength of an electron in its ground state?
To find the wavelength of an electron in its ground state in a hydrogen atom, would I or could I do the following?
Use the ground state energy (-13.6eV) in $E^2 = m^2c^4 + p^2c^2$
Solve for $p$
Use ...
0
votes
0answers
47 views
Finding the coefficients of a spinor
From the Schrödinger equation of a system I'm investigating, where the wave function is a 4-component spinor of coefficients $C_1, C_2, C_3, C_4$, I am able to obtain the expression
$\begin{pmatrix} ...
4
votes
1answer
109 views
Quantum Mechanics - Hidden Variables
In Steven Weinberg's Lecture on Quantum Mechanics (p. 342), he writes:
The correlation between the spins of the two particles can be
expressed as the average value of the product of the ...
3
votes
1answer
73 views
Tunneling and transmission
Lets say we have a tunelling problem in the picture, where $W_p$ is a finite potential step:
If particle is comming from the left a general solutions to the Schrödinger equations for sepparate ...
2
votes
1answer
98 views
Energy density of a quantum mechanical ensemble
How do we determine the energy density of a given system? I have seen that the density operator
$$\rho~=~\frac{\exp(-\beta \hat{H})}{\text{tr}(\exp(-\beta \hat{H}))}.$$
What does this mean exactly ...
0
votes
0answers
27 views
Circuit identities HTH [closed]
Using this circuit indetities $HXH=Z, HYH=-Y, HZH = X$ prove $HTH=R_x(\pi/4)$. here $H$ is Hadamard matrix, $X,Y$ and $Z$ are Pauli matrix, $R_x$ is a rotation matrix and $T=\left[ \begin{array}{cc}
1 ...
1
vote
1answer
70 views
Uncertainty Principle and Energy range for an electron in an atom
I have the following exercise:
Use Heisenberg's uncertainty principle and the relation $\Delta u = \sqrt{\langle u^2 \rangle - \langle u \rangle^2}$ to find the range of energy an electron has in an ...
2
votes
0answers
40 views
Analytical solution of two level system driving by a sinusoidal potential beyond rotating wave approximation
A quantum mechanical two-level system driving by a constant sinusoidal external potential is very useful in varies areas of physics. Although the wildly used rotating-wave approximation(RWA) is very ...
0
votes
0answers
43 views
Partial Measure Probability
Let be a
$$|\psi\rangle = \dfrac{3}{5\sqrt{2}}|00 \rangle- \dfrac{3i}{5\sqrt{2}}|01 \rangle+ \dfrac{2\sqrt{2}}{5}|10 \rangle - \dfrac{2\sqrt{2} i}{5}|11 \rangle$$
state with two qubits. ...
1
vote
1answer
62 views
Maximizing Multiplicity of Einstein Solid == (Temperature = $\infty$)?
If I have a system consisting of 2 Einstein solids (A and B) is it equivalent to say that maximizing the multiplicity of the ...
0
votes
0answers
68 views
Time-dependent perturbation theory [closed]
I am a student looking to understand the question given in the URL.
I understand how to complete earlier parts of this question. But the part I struggle with is figuring out which are the allowed and ...
0
votes
1answer
97 views
Potential step and its transmission / reflection
Lets say we have a potential step with regions 1 with zero potential $W_p\!=\!0$ (this is a free particle) and region 2 with potential $W_p$. Wave functions in this case are:
\begin{align}
...
1
vote
1answer
60 views
Why uncertainity is minimum for coherent states?
While reading for quantum damped harmonic oscillator, I came across coherent states, and I asked my prof about them and he said me it is the state at which $\Delta x\Delta y$ is minimum. I didn't ...
3
votes
1answer
92 views
Measurement and probability for quantum states
Suppose that the physical system is in generic state $|\psi\rangle$. Show that $\sum_{\lambda}p^2_{\lambda} = 1$ to an observable $O$, if and only if $\Delta O = 0$. ($\Delta O$ is a standard ...
1
vote
1answer
81 views
normalizing a wavefunction
I have a homework problem that I can't get started on, below is the first bit. I feel like I should just be able to integrate to find $C$ but I get a divergent integral. Can someone give me a hint as ...
0
votes
0answers
58 views
Quantum harmonic oscillator. Finding operators
Problem:
I'm trying to verify that $p_H(T)$ and $x_H(T)$ satisfy the following equations, (by solving the Heisenberg equation):
$x_H(t)=x_H(0)cos(\omega t)+(1/m\omega)p_H(0)sin(\omega t)$
...
0
votes
0answers
43 views
Wave equations for two intervals at Potential step
Lets say we have a potential step as in the picture:
In the region I there is a free particle with a wavefunction $\psi_I$ while in the region II the wave function will be $\psi_{II}$.
Let me ...
0
votes
0answers
33 views
Is it easier to determine the number of states with raising/lowering operators or using scattering?
A particle is bound by
$$V(x) = \begin{cases}\infty,& x <0 \\ \frac{-32\hbar^2}{ma}, & x\le a \\ 0, & x \le a\end{cases}$$
a) how many states are there?
i'm attempting ...
6
votes
2answers
146 views
Why does the quantum Heisenberg model become the classical one when $S\to\infty$?
The Hamiltonian of the spin $S$ quantum Heisenberg model is
$$H = J\sum_{<i,j>}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$$
I have read that when the spin quantum number $S\to\infty$, quantum fluctuation ...
3
votes
1answer
115 views
Bloch sphere representation
Suppose you know that a qubit is either is in state $|+\rangle$ with probability $p$ or in state $|-\rangle$ with probability $1-p$. If this is the best you know about the qubit's state, where in the ...
0
votes
1answer
96 views
Energies and numbers of bound states in finite potential well
Hello I understand how to approach finite potential well (I learned a lot in my other topic here). However i am disturbed by equation which describes number of states $N$ for a finite potential well (
...
0
votes
1answer
67 views
Why is the energy spectrum of bound QM plane wave continuous?
Please explain it in the context of this task: we have a potential barrier that looks like $\prod$, with $E<U$. There are 3 regions:
1) no field
2) barrier
3) no field
Solution could be ...
1
vote
0answers
61 views
linear response for a simple harmonic oscillator
Really sorry for this simple question, but I think it will be useful/interesting in general.
Consider a quantum simple harmonic oscillator.
Add a perturbation $H_I = -\lambda \hat{x}$
Calculate ...
2
votes
1answer
170 views
Show that for QM operator A: $\int_{-\infty}^{\infty}\psi A^{\dagger}A\psi dx = \int_{-\infty}^{\infty}(A\psi)^*(A\psi)dx $
I need to show for $$A = \frac{d}{dx} + \tanh x, \qquad A^{\dagger} = - \frac{d}{dx} + \tanh x,$$ that
$$\int_{-\infty}^{\infty}\psi^* A^{\dagger}A\psi dx = ...
2
votes
1answer
124 views
Coordinate representation of quantum ladder operator?
I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics.
I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
2
votes
2answers
280 views
Plotting $\psi$ for finite square well potential
Lets say we have a finite square potential well like below:
This well has a $\psi$ which we can combine with $\psi_I$, $\psi_{II}$ and $\psi_{III}$. I have been playing around and got expressions ...
