I am very interested in how Pauli found the Pauli matrices, so I read his original paper, but it didn't give me the perspective I wanted, so I went to Mehra and Rechenberg, but here's the thing, after reading Volumes 1, 2 and most of volume 3, I can't find any mention of Pauli matrices anywhere...
So on page 256 of Quantum Mechanics - The Theoretical Minimum, it says that the wave function of a momentum eigenvector, with respect to the position eigenbasis is ##\psi_p(x)=Ae^{\frac{ipx}{\hbar}}##, and ##A## must be ##\frac{1}{\sqrt{2\pi}}## to keep it a unit vector. However why must...
ATTEMPT AT SOLUTION: I understand if looking for positive this will be +hwo/2 (hbar) for Sz so must find |a|^2. and if looking for negative this will be -hwo/2 (hbar) so must find |b|^2. If asked to find say Sx and original question in Sz, we must find new eigenstates associated with this state...
Hi all
I am trying to follow a derivation of something involving second quantization formalism, I am stuck at this step :
$$
\sum_{m2}\sum_{\mu1}
\bra{2,m1,2,m2}\ket{k,q}\bra{2,\mu1,2,\mu2}\ket{k,-q}\delta_{-m2,\mu1}
= (-1)^{2+m2}\frac{\sqrt{2k+1}}{\sqrt{5}}\bra{k,-q,2,m2}\ket{2,-m1}\times...
Hi all
I was reading a certain paper that involves solving the Bohr-Mottelson Hamiltonian for a 5dimential square well potential, the B-M Hamiltoian reads:
my question is just how do I calculate the mass parameter "B" for a certain nuclei, and with a 5D infinite potential well how do I get the...
I am so new to circuit quantum electrodynamics. As far as I know, there are few things I could manipulate, like resonator, qubit resonance frequencies, Hamiltonian, coupling strength, Hilbert-space cutoff, dissipation rate, but they do not make sense to me and I do not how they can relate to my...
Hi,
I asked this question elsewhere, but I didn't understand the answer. It seems to be easy to understand, but for some reason I'm really confuse.
I'm not sure how to find the average position of an electron and the average separation of an electron and his proton in a hydrogen atom.
To be...
Knowing that to be Hermitian an operator ##\hat{Q} = \hat{Q}^{\dagger}##.
Thus, I'm trying to prove that ##<f|\hat{Q}|g> = <\hat{Q}f|g> ##.
However, I don't really know what to do with this expression.
##<f|\hat{Q}g> = \int_{-\infty}^{\infty} [f(x)^* \int_{-\infty}^{\infty} |x> <x| dx f(x)] dx##...
Hello.
I am looking to learn about averaging out a particle gas or any other type of organization of particles within a system or volume that can be approximated onto a grid or mesh where the particles are at a constant distance from each other: https://en.wikipedia.org/wiki/Particle_mesh.
I...
Hi,
I have hard time to really understand what's a stationary state for a wave function.
I know in a stationary state all observables are independent of time, but is the energy fix?
Is the particle has some momentum?
If a wave function oscillates between multiple energies does it means that the...
(0:00 / 0:42) photon going light-speed blender simulation
I have no idea how a mathematician would translate this example into an equation. Every time I've worked with soft bodies I seem to run short of mathematicians buddies. Regardless of the mathematics of continuous object deformation, this...
An electron beam with the spin state ## |\psi\rangle = \frac{1}{\sqrt{3}}|+\rangle+\sqrt{\frac{2}{3}}|-\rangle##, where ##\{|+\rangle,|-\rangle\}## is the eigenstates of ##\hat S_z##, passes through a Stern-Gerlach device with the magnetic field oriented in the ##Z## axis. Afterwards, it goes...
intermolecular distance means distance between particles. So, I imagine a sphere.
$$\frac{4}{3} \pi d^3 = \frac{V}{N}$$
However, Griffitfhs pictures a box instead, where
$$d^3 = \frac{V}{N}$$
And the difference between both models is a factor of ##(4\pi/3)^{2/5} \approx 1.8##, which is...
Hello everyone! I have two questions which had bothered me for quite some time. I am sorry if they are rather trivial.
The first is about the general solution of the hydrogen atom schrödinger-equation: We learned in our quantum mechanics class that the general solution of every quantum system...
I've been reading about Quantum Mechanics for years now and I think it's time I bought a textbook and really learned the math. I'm 15 y.o. and have a working understanding of Derivitives, Integrals and Vectors. Is this textbook a good one to start with or is it too complex? Which one would you...
On page 298 of Shankar's 'Principles of Quantum Mechanics' the author makes the statement :
""In an arbitrary ##\Omega## basis, ##\psi(\omega)## need not be even or odd, even if ##| \psi \rangle ## is a parity eigenstate. ""
Can anyone show me how this is the case when in the X basis...
Hello there, I am having trouble with part b. of this problem. I've solved part a. by calculating the commutator of the two observables and found it to be non-zero, which should mean that ##\hat B## and ##\hat C## do not have common eigenvectors. Although calculating the eigenvectors for each...
We know that thanks to the tunnel effect, in the case of a finite potential step (V) and considering a stationary state, when a plane wave with energy E < V encounter the step the probabability that the wave-particle coming from -∞ (where potential is V=0) will be ≠ 0, in particular the wave...
Suppose, there is an electro-optical modulator that can couple the neighboring modes in an optical ring resonator. The Hamiltonian for the system
looks something like this^^ (see the attached image). Here we sum over all modes m and 𝜙0 is a parameter. What will be a good set of basis for the...
hi guys
i was thinking about the inner product we choose in quantum mechanics to map the elements inside the hilbert space to real number which is given by :
$$\int^{∞}_{-∞}\psi^{*}\psi\;dV$$
or in some cases we might introduce a weight function dependent on the wave functions i have , it seems...
hi guys
i am trying to follow a proof of the generalized uncertainty principle and i am stuck at the last step :
i am not sure why he put these relations in (4.20) :
$$(\Delta\;C)^{2} = \bra{\psi}A^{2}\ket{\psi}$$
$$(\Delta\;D)^{2} = \bra{\psi}B^{2}\ket{\psi}$$
i tried to prove these using the...
What I have tried is a completing square in the Hamiltonian so that
$$\hat{H} = \frac{\hat{p}^2}{2} + \frac{(\hat{q}+\alpha(t))^2}{2} - \frac{(\alpha(t))^2}{2}$$
I treat ##t## is just a parameter and then I can construct the eigenfunctions and the energy eigenvalues by just referring to a...
hi guys
i am trying to solve the Asymptotic differential equation of the Quantum Harmonic oscillator using power series method and i am kinda stuck :
$$y'' = (x^{2}-ε)y$$
the asymptotic equation becomes :
$$y'' ≈ x^{2}y$$
using the power series method ##y(x) = \sum_{0}^{∞} a_{n}x^{n}## , this...
For the off-diagonal term, it is obvious that (p^2+q^2) returns 0 in the integration (##<m|p^2+q^2|n> = E<m|n> = 0##). However, (pq+qp) seems to give a complicated expression because of the complicated wavefunctions of a quantum harmonic oscillator. I wonder whether there is a good method to...
Starting with finding the probability of getting one of the states will make finding the other trivial, as the sum of their probabilities would be 1.
Some confusion came because I never represented the states ##|\pm \textbf{z}\rangle## as a superposition of other states, but I guess you would...
So let's say that we have som unitary matrix, ##S##. Let that unitary matrix be the scattering matrix in quantum mechanics or the "S-matrix".
Now we all know that it can be defined in the following way:
$$\psi(x) = Ae^{ipx} + Be^{-ipx}, x<<0$$ and $$ \psi(x) = Ce^{ipx} + De^{-ipx}$$.
Now, A and...
Specifically, for this section/year: https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/assignments/.
I ask for those problem sets because I am following Prof. Barton Zwiebach's lectures on edX and the website doesn't seem to parse the HTML for the assignments always. What...
Once I know the Hamiltonian, I know to take the determinant ##\left| \vec H-\lambda \vec I \right| = 0 ## and solve for ##\lambda## which are the eigenvalues/eigenenergies.
My problem is, I'm unsure how to formulate the Hamiltonian. Is my potential ##U(r)## my scalar field ##\phi##? I've seen...
Hi, so I'm having trouble with a homework problem where it asks me to find the number of states with an energy less than some given E.
From this, I was able to work out the energy E to be
$$
E = \frac{\hbar^2}{2m} \frac{\pi^2}{a^2} \left(
n_x^2 + n_y^2 + n_z^2
\right)
$$
and...
Some questions:
Why is this even a valid wave function? I thought that a wave function had to approach zero as x goes to +/- infinity in all of space. Unless all of space just means the bounds of the square well.
Since we have no complex components. I am guessing that the ##\psi *=\psi##.
If...
Hello!
I have checked commutator matrix form of $$\vec{p}=im/\hbar [H,\vec{x}]$$ but I realized i don't undertand something
I have $$[H,\vec{x}]=H\vec{x}-\vec{x}H$$ and
$$(H\vec{x})_{i}=H_{ij}x_j$$ & $$\ ( \vec{x}H)_{i}=x_jH_{ji}$$
but what is the second term matrix representation...
I've been reading about Quantum Field Theory and what it says about subatomic particles. I've read that QFT regards particles as excited states of underlying quantum fields.
If this is the case, how can particles be regarded as objective? It seems to me that this also removes some of the...
Attempt: I'm sure I know how to do this the long way using the definition of stationary states(##\psi_n(x)=\sqrt{\frac {2} {a}} ~~ sin(\frac {n\pi x} {a})## and ##\int_0^{{a/2}} {\frac {2} {a}}(1/5)\left[~ \left(2sin(\frac {\pi x} {a})+i~ sin(\frac {3\pi x} {a})\right)\left( 2sin(\frac {\pi x}...
I need help with part d of this problem. I believe I completed the rest correctly, but am including them for context
(a)Show that the hermitian conjugate of the hermitian conjugate of any operator ##\hat A## is itself, i.e. ##(\hat A^\dagger)^\dagger##
(b)Consider an arbitrary operator ##\hat...
Hi everyone, was just wondering what people think is a good undergraduate QM book is as opposed to Griffiths. I've read through it, and I have looked and many people say it is good for people who've never been exposed to QM before, but when it comes to solving problems I struggle a lot, and...
I worked out the expectation values of the components of a 1/2 spin particle. However, I'm confused about Griffiths notation for the x and y components.
For the x component I got, ## \left< S_x \right> = \frac \hbar 2 (b^*a+a^*b)## which is correct, but Griffiths equates this to ##...
If I calculate ## <\psi^0|\epsilon|\psi^0>## and ## <\psi^0|-\epsilon|\psi^0>## separately and then add, the correction seems to be 0 since ##\epsilon## is a constant perturbation term.
SO how should I approach this? And how the Δ is relevant in this calculation?
The wavefunction is Ψ(x,t) ----> Ψ(λx,t)
What are the effects on <T> (av Kinetic energy) and V (potential energy) in terms of λ?
From ## \frac {h^2}{2m} \frac {\partial^2\psi(x,t)}{\partial x^2} + V(x,t)\psi(x,t)=E\psi(x,t) ##
if we replace x by ## \lambda x ## then it becomes ## \frac...
Summary: In the past, physicists talked of the phenomenon of "wave function collapse" very freely, whereas now there seems to be some reservation about it. Why?
In reading older popular physics literature, physicists used to talk about "wave function collapse" freely and more often...
I am an undergraduate doing research on QC/QI. My current topic to learn is continuous-time quantum walks, but first I must learn the random quantum walk. That being said, I was wondering if someone could simply explain what a random quantum walk is and then explain how they could be useful with...
I've been reading Griffths' intro to elementary particles and I encountered this symbol that looks similar to "M" called amplitude, which can be calculated by analyzing the Feynman diagram of an interaction. What exactly is it? When I hear amplitude I imagine waves, but not sure what this one's...
If it is the asymptotic behavior of the Airy's function what it's used instead of the function itself: Does it mean that the wkb method is only valid for potentials where the regions where ##E<V## and ##E>V## are "wide"?