0
votes
1answer
75 views

Showing Dirac equation's Lorentz invariance and use of unitary matrix $U$

Dirac equation is $i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0 $ To show its Lorentz invariance, we convert spacetime into $x'$ and $t'$ from $x$ and $t$ and then $( iU^\dagger \gamma^\mu ...
2
votes
1answer
92 views

In the Dirac equation, do $\alpha$ and $p$ commute?

The Dirac Hamiltonian is given as $H = \vec \alpha·\vec pc + \beta mc^2$ , Do the alpha and beta operators commute with the momentum operator? If yes then how?
2
votes
3answers
244 views

Dirac equation in QFT vs relativistic QM

How does the Dirac equation in quantum field theory solve the existing problems in the interpretation Dirac equation (as a single-particle wave equation) in relativistic quantum mechanics? EDIT: The ...
7
votes
2answers
211 views

Why the lowest order of matrices in Dirac equation are 4x4 matrices?

Why the lowest order of matrices in Dirac equation (Relativistic Quantums) are 4x4 matrices (and can not be 2x2 matrices)? How to prove it?
1
vote
2answers
159 views

Which of these two different forms of spin-orbit interaction is correct?

I am seeing the spin-orbit interaction in two different ways: $\lambda [\mathbf{p} \times \nabla V]\cdot \sigma$ $\lambda [\nabla V \times \mathbf{p}]\cdot \sigma$ I don't see how these two ...
3
votes
2answers
202 views

Wavefunction of an electron

Electron is a spin $\frac{1}{2}$ particle, so needs 2-component wave function but the Dirac theory of electron is based on 4-component wave function, are all components of it independent or only two ...
5
votes
2answers
299 views

Is the Dirac Lagrangian Hermitian?

I'm wondering of the Dirac Lagrangian density $$\mathcal{L} =\overline{\psi}(-i\gamma^\mu \partial_\mu +m)\psi $$ is an hermitian operator, since upon complex conjugating one gets ...
4
votes
1answer
178 views

Derivation of the quadratic form of the Dirac equation

I am asked to derive the quadratic form of the Dirac equation in an electromagnetic field, $\left[\left(i\hbar \partial - \frac{e}{c}A\right)^2 - \frac{\hbar e}{2c} \sigma^{\mu\nu} F_{\mu\nu} - ...
2
votes
1answer
188 views

Massless spin 1/2 particle

Could a massless spin 1/2 particle, or more generally massless half-integer spin particles exist? Does it make sense to say that they could be described for example by the Dirac equation by forgetting ...
3
votes
1answer
107 views

Negative energy solutions Dirac equation without radation field

In the book "Relativistic Quantum Mechanics" by Bjorken and Drell in Chapter 5.1 page 64 there is the following statement about the problem of negative solutions to the Dirac equation: By their ...
1
vote
1answer
316 views

Dirac field and stress-energy tensor density

I read somewhere that stress-energy tensor density is a symmetric tensor. But if I take the Dirac Field tensor: $$T^{\mu \nu}=i \psi^\dagger \gamma^0 \gamma^\mu \partial^\nu \psi $$ How could I ...
0
votes
1answer
125 views

Explicit solutions to 2-d Dirac Equation

The 2-d Dirac equation without any constants is represented usually as $$i*dt (\phi) = D (\phi)$$ where $D = m\sigma_2-i\sigma_1dx-i\sigma_3dy$. Where can I find explicit closed form solutions to ...
0
votes
1answer
284 views

Showing Dirac Hamiltonian is hermitian

I'm trying to show that $H_D = -i\boldsymbol{\alpha}.\nabla+\beta m$ is hermitian. Its given that $$ \gamma^{0\dagger}=\gamma^0 $$ $$ \boldsymbol\gamma^\dagger=-\boldsymbol\gamma $$ What i've done ...
6
votes
4answers
475 views

Why is the Dirac equation not used for calculations?

From what I understand the Dirac equation is supposed to be an improvement on the Schrödinger equation in that it is consistent with relativity theory. Yet all methods I have encountered for doing ...
4
votes
0answers
285 views

Relativistic genarization of Quantum Harmonic Oscillator

I am trying to find out relativistic description of a quantum harmonic oscillator. For a classical relativistic oscillator mass is a function of co-ordinates(http://arxiv.org/abs/1209.2876). ...
10
votes
2answers
684 views

Dimension of Dirac $\gamma$ matrices

While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices: ...
1
vote
0answers
211 views

Matrix manipulation for Dirac matrices

From the Dirac equation in gamma matrices, we know that $$\gamma^i=\begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix}$$ and $$\gamma^0=\begin{pmatrix} I & 0 \\ 0 & -I ...
3
votes
2answers
321 views

Matrix operation in dirac matrices

If we define $\alpha_i$ and $\beta$ as Dirac matrices which satisfy all of the conditions of spin 1/2 particles , p defines the momentum of the particle, then how can we get the matrix form ? ...
5
votes
1answer
809 views

What is the relativistic particle in a box?

I know people try to solve Dirac equation in a box. Some claim it cannot be done. Some claim that they had found the solution, I have seen three and they are all different and bizarre. But my main ...
1
vote
2answers
149 views
3
votes
2answers
298 views

Energy spectrum of a Dirac electron

How do you explain easily "The spectrum of an electron in a repulsive potential " and hence "bound state of charge conjugation" in Dirac hole theory ?
0
votes
1answer
583 views

Charge conjugation in Dirac equation

I need to know the mathematical argument that how the relation is true $(C^{-1})^T\gamma ^ \mu C^T = - \gamma ^{\mu T} $ . Where $C$ is defined by $U=C \gamma^0$ ; $U$= non singular matrix , $T$= ...
3
votes
1answer
421 views

Matrix order in Dirac equations

The trace of matrix is always sum of its eigen values , which can be seen if $\hat{U}$ transforms the matrix $\alpha_i$ into it's diagonal form . $$ \begin{pmatrix} A_1 & 0 & \cdots & 0 ...
3
votes
4answers
1k views

Why would Klein-Gordon describe spin-0 scalar field while Dirac describe spin-1/2?

The derivation of both Klein-Gordon equation and Dirac equation is due the need of quantum mechanics (or to say more correctly, quantum field theory) to adhere to special relativity. However, excpet ...
0
votes
1answer
492 views

Solution to Klein-Gordon equation always valid?

We know that there is a relativistic version of Schrodinger equation called Klein-Gordon equation. However, it has some problems and due to these problems, there is Dirac equation that handles these ...
1
vote
1answer
249 views

Complete set and Klein-Gordon equation

In http://www.physics.ucdavis.edu/~cheng/teaching/230A-s07/rqm2_rev.pdf, it says that when there is some external potential, the Klein-Gordon equation is altered, and it says the following: The ...
2
votes
1answer
391 views

Explanation of equation that shows a failed approach to relativize Schrodinger equation

I'm reading the Wikipedia page for the Dirac equation: $\rho=\phi^*\phi\,$ ...... $J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$ with the conservation of probability ...
1
vote
1answer
514 views

How did one get the defining equation of probability current and conservation of probability current and density?

I'm reading the Wikipedia page for the Dirac equation: $$\rho=\phi^*\phi$$ and this density is convected according to the probability current vector $$J = ...
2
votes
2answers
767 views

How to obtain Dirac equation from Schrodinger equation and special relativity?

I'm reading the Wikipedia page for the Dirac equation: The Dirac equation is superficially similar to the Schrödinger equation for a free massive particle: A) ...
2
votes
1answer
545 views

How to construct the charge conjugation matrix for any given dimension?

Generally, Gamma matrices could be constructed based on the Clifford algebra. \begin{equation} \gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij}, \end{equation} My question is how to generally ...
2
votes
1answer
336 views

Magnetic moment derivation from Dirac equation

I am reading a text book where they show the electron has spin 1/2 using Dirac's equation. At one point in the derivation they define $\pi=P-qA/c$ where $P$ is the momentum operator and A is the ...
0
votes
0answers
82 views

Is it necessary to use all solutions when calculating an expectation value in a spin state?

I'm given an spinor $\Psi$ which is solution of the Free Dirac equation, such that is an eigenfunction of $\hat{\vec{p}}$ and has positive energy. Then I'm asked to calculate the expectation value of ...
3
votes
2answers
540 views

Spin matrices in Dirac equation

Why in every textbook when deriving Dirac's equation the smallest possible matrices ($2 \times 2$) are used? I wonder why one couldn't use spin 1 matrices ($3 \times 3$) and get relativistic equation ...
4
votes
4answers
1k views

Where is spin in the Schroedinger equation of an electron in the hydrogen atom?

In my current quantum mechanics, course, we have derived in full (I believe?) the wave equations for the time-independent stationary states of the hydrogen atom. We are told that the Pauli Exclusion ...
5
votes
4answers
1k views

Why are four-vectors needed in the Dirac equation, when there are 4 linearly independent 2D matrices?

I was taught that for the Dirac-equation to "work", you need matrices of the following form: $Tr(\alpha^i) = 0$. Eigenvalues +1 or -1 2 previous points together: equal number of negative and ...
7
votes
4answers
839 views

Dirac equation on general geometries?

I have a numerical method for computing solutions to the Dirac equation for a spin 1/2 particle constrained to an arbitrary surface and am interested in finding applications where the configuration ...
5
votes
7answers
901 views

Evolution in the interpretation of the Dirac equation

As I understand, Dirac equation was first interpreted as a wave equation following the ideas of non relativistic quantum mechanics, but this lead to different problems. The equation was then ...