In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of 1/2. The spin number describes how many symmetrical facets a particle has in one full rotation; a spin of 1/2 means that the particle must be rotated by two full turns (through 720°) before it has the same configuration as when it started.
Particles having net spin 1/2 include the proton, neutron, electron, neutrino, and quarks. The dynamics of spin-1/2 objects cannot be accurately described using classical physics; they are among the simplest systems which require quantum mechanics to describe them. As such, the study of the behavior of spin-1/2 systems forms a central part of quantum mechanics.
Could anyone help with some of the later parts of the derivation for Dirac spinors, please?
I understand that an arbitrary vector ##\vec v##
$$ \begin{bmatrix}
x \\
y \\
z
\end{bmatrix} $$
can be defined as an equivalent matrix V with the components
$$ \begin{bmatrix}
z & x - iy \\
x + iy...
According to Chapter 8 of Griffiths' book Introduction to Electrodynamics, the magnetization force that acts on a magnetic dipole is
$$F_M=\nabla (m \cdot B)$$,
where ##m## is the magnetic moment and ##B## is the magnetic field.
For a paramagnetic or diamagnetic particle...
So I understand that fermions are anti-symmetric under exchange, but in the contexts I've seen this explained they were always talking about two particles, or at least two wavefunctions. I'm curious how this works when there are three or more particles. Is any two given pairs of those 3+...
Hi,
While studying the spin 1/2, I'm facing some confusions about the spinors and the eigenspinors.
I understand that ##\chi = \begin{bmatrix}a \\ b \end{bmatrix}## is the spinor with ##\chi_+ = \begin{bmatrix}1 \\ 0 \end{bmatrix}## and ##\chi_-= \begin{bmatrix}0 \\ 1 \end{bmatrix}## the...
When the expectation value of spin in the z direction for one particle is zero and I make measurements for an even number of particles in the same state, do I get exactly half to be spin up and half to be spin down along the z direction? More generally, what does spin expectation value for one...
What's the difference between a bra vector and ket vector in specifying spin states except for notational convenience when calculating probablility amplitudes? Are they equivalent?
If I want to get the spin angular momentum of a particle using the Stem-Gerlach experiment, I think I will find the spin 1/2 particle either spin up or spin down, but not both. I however want to ask this : Is there a non-zero probability that a particle which is spin-up in the z direction to be...
Let's take a beam of spin 1/2 particles prepared in the state |up> in the Z direction, let's pass it through a Stern& Gerlach apparatus in the X direction to get two beams of spin |right> and spin |left>, and then redirect these two beams directly in another inverted S&G to reunite them in one...
I know that we can change the spin orientation of a spin 1/2 particle up or down and test it in the Stern Gerlach apparatus.
And the spin 1/2 particles need two full rotations to return to the previous state.
Questions:
1). what does state mean?
2). Is, Changing spin orientation to up or...
The other day I found a fascinating video on geometric algebra:
At 34:50, after showing how to rotate a vector in three dimensions, he says, "wait a minute, this looks like a spinor from quantum mechanics. The way that spinors rotate is always said to be a part of so-called 'quantum...
So what I'm not sure on, is calculating the matrix elements for part (iii) with Pauli spinors and Pauli matrices, and then finding the form of the corresponding states. As I don't see how using the hint helps.
The following is using the eigenvalues of the spin-operators.
Provided what I...
I am having trouble to understand what it means by "physically relevant real parameters" and how does it help us to specify a quantum system.
Let say, we have a state of k half spin electrons? My guess is about the local phase of the spin, and this would make it 2^k parameters since each...
Well, this calculation is straightforward in the Heisenberg picture. After finding the eigen values and eigen vectors of the total Hamiltonian, I found the explicit form for the exponential of the integral of the matrix and then did the matrix multiplication and calculated its expectation value...
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...
Hello,
I have a question about a statement made on a YouTube physics lecture
I was (am) working through chapter 4 section 4 (4.4) - “Spin” of Griffiths. (only because I own this book ) I found the YouTube lectures by searching for phrases like “quantum Griffiths online lectures”. One of the...
For a tetrahedron with four spin (1/2) particles, I know there are three separate energy levels at $$l=2,l=1,and l=0$$. My question is how I would go about finding the degeneracy of each level. I know that the number of states must be $$2^4$$. Any clues on where to start would be appreciated...
Summary: please help me understand the following questions from Physics GRE test Thank you very much! To be honest i really hate this formalism. Memorizing such things is pain. but like everything it is what it is
Hello,
I'm trying to couple 3 spin 1/2 particles. So far, I have been able to find the coefficient for the other states but I can't get the results for ##j_{12} = 0## to ##j_3=1/2##.
Here is my attempt:
1) Using CG table...
Why does the derivation of the Dirac equation naturally lead to spin ½ particles? The equation is derived from very general starting assumptions, so which of these assumptions has to be wrong to give us a spin-0 or spin-1 particle?
I have tried to search for an answer and got as far as this...
I am studying identical particles and I have some doubts. Considerer two identical spin 1/2 particles interacting through a central potential ##V##. In the rest of CM, the hamiltonian is $$ H = \frac{\textbf{P}^{2}}{2M} + \frac{\textbf{p}^{2}}{2\mu} + V(r),$$ where ##\textbf{P}## is the momentum...
I am studying the deuterium's nucleus.
As we know, there are just two eigenstates for a spin 1/2 particle: either spin up or spin down.
Thus, over the whole nucleus, you get 4 possible combinations:
1) Spin up-spin up
2) Spin up-spin down
3) Spin down-spin up
4) Spin down-spin down
If you...
Homework Statement
Given the expression
s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1>
obtain the matrix representations of s+/- for spin 1/2 in the usual basis of eigenstates of sz
Homework Equations
s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1>
S_{+} = \hbar...
In the paper "Steady-state spin synchronization through the collective motion of trapped ions" it states the following:
"Steady-state synchronization of atomic dipoles forms the foundation for ultra-stable optical lasers utilizing
narrow-linewidth atoms coupled to a lossy cavity mode. The...
Homework Statement
The ground state energy of 5 identical spin 1/2 particles which are subject to a one dimensional simple harkonic oscillator potential of frequency ω is
(A) (15/2) ħω
(B) (13/2) ħω
(C) (1/2) ħω
(D) 5ħω
Homework Equations
Energy of a simple harmonic oscillator potential is
En...
Homework Statement
Two spin ##\frac 12## particles form a composite system. Spin A is in the eigenstate ##s_z = + \frac 12## and Spin B is in the eigenstate ##s_x = + \frac 12## What is the probability that a measurement of the total spin will give the value ##0##?
Homework Equations
I know...
Hey everybody,
I am trying to expand a system of seven qubits from one dimensional Hamiltonian to the two dimensional representation.
I have the one dimensional representation and I don't know what to add to transform it from 1D to 2D representation.
I would really appreciate your help and...
A spin 1/2 particle is represented by a spinor while its position is represented by a three-vector. What object should we use to represent such particle if we want to consider both features? That is, what object should we use if we want to consider both spin and space position?
It seems there's...
Hey,
I am studying Spin 1/2 system, the case of 2 electrons in a magnetic field, since we have 2 electrons, we expect that the matrix will be of the size 4x4, which is what I have got.
As I know that I could use the density matrix to calculate the expectation values of any physical quantity...
As I understand it (e.g. from discussions around the Fermi field theory of the nuclear force), a spin 1/2 particle can emit a spin-1 particle and simultaneously flip its spin (say, spin +1/2 -> photon +1 & spin -1/2); but how does this work with spin-2 particles? Does it need to emit pairs in...
Suppose we confine a spin 1/2 particle to an infinite annular region, in cylindrical coordinates, defined by the two cylinders r=a and r=b with a<b. How does such a region constrain possible spin and angular momentum?
Thanks!
Homework Statement
The problem is given above. I am struggling on the first two parts, where I am tasked with finding gamma and beta.
Homework Equations
For spin one half states in arbitrary directions, I know that psi = a*|n;+> + b*|n;-> .
|n;+> = cos(theta/2)|+> +sin(theta/2)*ei*phi|->...
Hi buddies.
I recently finished my quantum mechanics course, however, I would like to know the solution of this exercise because i couldn´t solve it on my last exam, and i would like to take this doubt off.
An operator ##F## describing the interaction of two spin ##\frac{1}{2}## particles has...
Homework Statement
Homework Equations
This is a passage from Modern Quantum Mechanics by Sakurai ( page 26~27)
The Attempt at a Solution
I wonder how i can get 1.4.8 , 1.4.9 equations . and what do they mean?
Homework Statement
Show that for a two spin 1/2 particle system, the expectation value is \langle S_{z1} S_{n2} \rangle = -\frac{\hbar^2}{4}\cos \theta when the system is prepared to be in the singlet state...
Homework Statement
Two particles, their spin are 1/2.
The hamiltonian is ##H=\gamma s_1 \cdot s_2##
At t=0, the state ##|\alpha(0)>## is such as ##s_{1z}|\alpha(0)>=\hbar/2 |\alpha(0)>## and ##s_{2z}|\alpha(0)>=\hbar/2 |\alpha(0)>##. Find the state ##|\alpha(0)>##.2. The attempt at a...
We know Sz = h/2 [1 0 // 0 -1] for up spin.So can we say that the energy of the electron in this state is h/2 ?
(since Sz |up> = h/2 |up>)Now we consider a Hamiltonian H = [1 0 // 0 -1] which has eigen values +1, -1 and the respective states are
(1 0) and (0 1).
If we see the...
In my book (Griffiths) it is said :
A particle is in ↑ state.
The z-component of the particle's spin angular momentum [Sz] is h bar/2.
The x- component of the particle's spin angular momentum [Sx] can be either h bar/2 or...
Hi,
I have learned about how to find the 4 spin states of 2 spin 1/2 particles, and how to find them by using the lowering operator twice on |1/2, 1/2> to find the triplet, then simply finding the orthogonal singlet state, |0, 0>.
I started to attempt finding the states of 3 spin 1/2...
Homework Statement
So I'm given a spin 1/2 particle in a rotating magnetic field in the (x,y) direction and a constant field, B_0, in the z direction and am asked to find the S matrix describing it. Given is:
B(t) = [B_1 \cos(\omega t), B_1 \sin(\omega t), B_0]
Homework Equations
H = \sum...
Hi,
I was looking at the wikipedia article for spin 1/2 http://en.wikipedia.org/wiki/Spin-%C2%BD and it stated that √(½(½+1))ħ=√3/2ħ. According to my math it is √¾ħ. Is this just a typo or am I missing something?
The spin observable for spin 1/2 particles is represented by Pauli Matrices acting upon a 2-dimensional Hilbert Space. In RQM, forgetting about the matter-antimatter duality for the moment, that TWO-state Hilbert Space is directly related, through the Lorentz Group, to the TWO separate...
Hi everyone!
I am trying to create the density matrix for a spin-1/2 particle that is in thermal equilibrium at temperature T, and in a constant magnetic field oriented in the x-direction. This is a fairly straightforward process, but I'm getting stuck on one little part.
Before starting I...
I have only ever seen the wavefunction for a spin 1/2 particle written in the basis set |α> |β>. I was interested in how a wavefunction |ψ> = a|α> + b|β> might be rewritten in a continuous basis and hence would need to know what the actual functions of |α> & |β> were.
Thanks
hello guys , in this problem from zettili quantum mechanics that i attach , i think something is wrong , first the problem said two particles with spin 1/2 but didn't mention that the system is in singlet state or triplet state , so if the system be in triplet state then our spatial wave...
Homework Statement
For a spin 1/2 system, the eigenstates of z-component of the angular-momentum operator Sz are given by:
S_z |\pm> = \pm \frac{\hbar}{2}|\pm>
Suppose at time t, the state of the system is given by:
|\psi> = a|+> + b|->
If Sz is measured, what are the possible results of...
So I know this might be a lot to read but I am having a very hard time understanding how to use the formulas in degenerate perturbation theory. Here is the problem I am on.
Homework Statement
A system of two spin-1/2 particles is described by the following Hamiltonian...
Hi,
I am having trouble following the Peskin and Schroeder and their derivations to show that a Dirac particle is a spin 1/2 particle (page 60 and 61). I understand how he gets the first (unnumbered) equation on page 61. However, I don't understand how he gets to the second equation...
Homework Statement
A beam of spin-1/2 particles is prepared in the state
\left|\psi\right\rangle=\frac{2}{\sqrt{13}}\left|+\right\rangle_x+i \frac{2}{\sqrt{13}}\left|-\right\rangle_x
￼￼￼￼a) What are the possible results of a measurement of the spin component Sz, and with what probabilities...