The dirac-equation tag has no wiki summary.
10
votes
1answer
495 views
Exact energies of spherical harmonic oscillator in Dirac equation
The potential is given by:
$$
V(r) = {1\over 2} \omega^2 r^2
$$
and we are solving the radial Dirac equation (in atomic units):
$$
c{d P(r)\over d r} + c {\kappa\over r} P(r) + Q(r) (V(r)-2mc^2) = E ...
9
votes
1answer
297 views
Dimension of Dirac $\gamma$ matrices
While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices:
...
9
votes
1answer
213 views
How is the Dirac adjoint generalized?
I am wondering how one can generalize the Dirac adjoint to flat "spacetimes" of arbitrary dimension and signature. To be more specific, a standard situation would be to consider 4 dimensional ...
8
votes
2answers
299 views
Dirac equation as canonical quantization?
First of all, I'm not a physicist, I'm mathematics phd student, but I have one elementary physical question and was not able to find answer in standard textbooks.
Motivation is quite simple: let me ...
7
votes
3answers
451 views
What was missing in Dirac's argument to come up with the modern interpretation of the positron?
When Dirac found his equation for the electron $(-i\gamma^\mu\partial_\mu+m)\psi=0$ he famously discovered that it had negative energy solutions. In order to solve the problem of the stability of the ...
7
votes
4answers
682 views
Dirac equation on general geometries?
I have a numerical method for computing solutions to the Dirac equation for a spin 1/2 particle constrained to an arbitrary surface and am interested in finding applications where the configuration ...
6
votes
2answers
266 views
Dirac equation in curved space-time
I have seen the Dirac equation in curved space-time written as $$[i\bar{\gamma}^{\mu}\frac{\partial}{\partial x^{\mu}}-i\bar{\gamma}^{\mu}\Gamma_{\mu}-m]\psi=0 $$
This ...
6
votes
2answers
381 views
Charge conjugation in Dirac equation
According to Dirac equation we can write,
\begin{equation}
\left(i\gamma^\mu( \partial_\mu +ie A_\mu)- m \right)\psi(x,t) = 0
\end{equation}
We seek an equation where $e\rightarrow -e $ and which ...
6
votes
3answers
211 views
Why is the Dirac equation not used for calculations?
From what I understand the Dirac equation is supposed to be an improvement on the Schrödinger equation in that it is consistent with relativity theory. Yet all methods I have encountered for doing ...
6
votes
3answers
563 views
What is the difference between a spinor and a vector or a tensor?
Why do we call a 1/2 spin particle satisfying the Dirac equation a spinor, and not a vector or a tensor?
5
votes
7answers
762 views
Evolution in the interpretation of the Dirac equation
As I understand, Dirac equation was first interpreted as a wave equation following the ideas of non relativistic quantum mechanics, but this lead to different problems.
The equation was then ...
5
votes
4answers
786 views
Why are four-vectors needed in the Dirac equation, when there are 4 linearly independent 2D matrices?
I was taught that for the Dirac-equation to "work", you need matrices of the following form:
$Tr(\alpha^i) = 0$.
Eigenvalues +1 or -1
2 previous points together: equal number of negative and ...
5
votes
1answer
147 views
Higher dimension operator in free Dirac Lagrangian
When discussing higher dimensional operators in a theory with fermions, why do I never see anyone ever talk about the dimension five operator $\partial_\mu\bar\psi\partial^\mu\psi$?
How does the ...
4
votes
2answers
225 views
Relation for Dirac-spinors of different helicities
Assume that we have massless spin-1/2 particles. The Dirac-spinor is the solution of the Dirac equation:
$$ p^\mu \gamma_\mu u_\pm(p) = 0, \quad p^2 = 0$$
The subscripts $\pm$ denote two different ...
4
votes
2answers
566 views
What is negative about negative energy states in the Dirac equation?
This question is a follow up to What was missing in Dirac's argument to come up with the modern interpretation of the positron?
There still is some confusion in my mind about the so-called ...
4
votes
1answer
388 views
Is Zitterbewegung an artefact of single-particle theory?
I have seen a number of articles on Zitterbewegung claiming searches for it such as this one: http://arxiv.org/abs/0810.2186. Others such as the so-called ZBW interpretation by Hestenes seemingly ...
4
votes
3answers
258 views
How does one interpret the Dirac equation with a self-field potential?
EVERY QFT text I've ever examined states that if there is an external vector potential, $A_\mu$, then one writes the Dirac eq.(or Klein-Gordon eq.) using a covariant derivative to include this U(1) ...
4
votes
1answer
455 views
What is the relativistic particle in a box?
I know people try to solve Dirac equation in a box. Some claim it cannot be done. Some claim that they had found the solution, I have seen three and they are all different and bizarre. But my main ...
3
votes
4answers
675 views
Where is spin in the Schroedinger equation of an electron in the hydrogen atom?
In my current quantum mechanics, course, we have derived in full (I believe?) the wave equations for the time-independent stationary states of the hydrogen atom.
We are told that the Pauli Exclusion ...
3
votes
2answers
352 views
Lorentz transformations in Dirac equation
Let's denote a spinor $\xi$. If $(\theta ,\phi)$ are the parameters of a rotation and pure Lorentz transformation, then how $\xi$ could be written as
$$\xi ~\rightarrow~ \exp\left(\ i ...
3
votes
1answer
158 views
What happens to the Lagrangian of the Dirac theory under charge conjugation?
Consider a charge conjugation operator which acts on the Dirac field($\psi$) as
$$\psi_{C} \equiv \mathcal{C}\psi\mathcal{C}^{-1} = C\gamma_{0}^{T}\psi^{*}$$
Just as we can operate the parity operator ...
3
votes
1answer
130 views
Charge and the Dirac field
In Zee's quantum field theory in a nutshell, 2nd edition, pg 550 he has
$Q=\int {d^3p \over (2\pi)^3(E_p/m)} \sum_s \{b^\dagger(p,s)b(p,s)-d^\dagger(p,s)d(p,s)\}$
showing clearly that $b$ ...
3
votes
1answer
171 views
Does Dirac's idea of filled negative energy states make sense?
Please bear with me a bit on this. I know my title is controversial, but it's serious and detailed question about the explanation Dirac attached to his amazing equations, not the equations themselves.
...
3
votes
2answers
200 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 ?
...
3
votes
1answer
203 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 ?
3
votes
0answers
104 views
Dirac action and conventions
I have a (possibly) fundamental question, which is driving me crazy.
Notation
When considering the Dirac action (say reading Peskin's book), one have
$\int ...
2
votes
2answers
86 views
Sign convention for basic Dirac equation
The dirac equation;$$(i\gamma^\mu\partial_{\mu} - m)\psi=0 $$ is just; $$(i\gamma^{0}\partial_{0} - i\gamma^{i}\partial_{i} - m)\psi=0 $$ in a (+,---) metric right?
2
votes
4answers
480 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 ...
2
votes
3answers
318 views
Dirac equation as Hamiltonian system
Let us consider Dirac equation
$$(i\gamma^\mu\partial_\mu -m)\psi =0$$
as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that ...
2
votes
1answer
212 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 ...
2
votes
1answer
148 views
Dirac trace theorem
I am unable to prove exactly one trace identity that appears in the appendix of Peskin and Schroeder's QFT book. Can someone help me?
The theorem [Appendix A.4 eqn (A.28)] says that the order of ...
2
votes
2answers
132 views
numerical formulation of Dirac equation plus electromagnetic field
I have the following equations describing the electron field in a (classic) electromagnetic field:
$$ c\left(\alpha _i\right.{\cdot (P - q(A + A_b) + \beta mc) \psi = E \psi } $$
where $A_b$ is ...
2
votes
1answer
246 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
397 views
What is the exponential form gamma matrix for a general rotation and boost?
It would be nice to have a cute method that uses Lorentz transformations of basis vectors by exponential transformation using gamma matrices. To avoid confusion, let's assume -+++ signature. Given ...
2
votes
1answer
484 views
Dirac equation algebraic derivation, a gauge symmetry
Suppose i try to derive the most generic Dirac-like equation (that is, as factors of first-order expression in momenta and mass operator where we allow coefficients that are associative, don't ...
2
votes
1answer
221 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 ...
2
votes
2answers
151 views
Finding wave-fuctions of a Dirac particle for given 4-momentum and spin 4-vector
I've been reading through various materials on relativistic quantum mechanics, but I find the lack of simple examples disturbing.
I'm acquainted with the general form the solutions to the Dirac ...
2
votes
0answers
61 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).
...
1
vote
3answers
105 views
Problem involving Dirac's equation
I'm stuck in an equation derivation of Ryder's QFT book.
Starting with Dirac's equation:
$$(i\gamma^\mu\partial_\mu-m)\psi=0$$
If I multiply by $i\gamma^\nu\partial_\nu$, I get:
...
1
vote
2answers
106 views
A step in the derivation of the magnetic momentum of the electron in Zee's QFT book
In chapter III.6 of his Quantum Field Theory in a Nutshell, A. Zee sets out
to derive the magnetic moment of an electron in quantum electrodynamics.
He starts by replacing in the Dirac equation the ...
1
vote
2answers
187 views
momentum four vector and dirac matrices
$$c\left(\alpha _i\right.{\cdot P + \beta mc) \psi = E \psi } $$
From the above dirac equation it can be shown for zero momenta that spin and antimatter are associated with $\beta $.
On the other ...
1
vote
2answers
327 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 ...
1
vote
2answers
206 views
Geometrical interpretation of the Dirac equation
Is there a geometrical intuitive picture behind the Dirac equation, and the gamma matrices that it uses? I know the geometric algebra is a Clifford algebra. Can the properties of geometric algebra, be ...
1
vote
2answers
99 views
1
vote
1answer
162 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 ...
1
vote
1answer
227 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) ...
1
vote
0answers
40 views
Lagrangians for non-local equations of motion
Say I have a multicomponent field $X_a(x,t)$ such that I know it Fourier modes satisfy the following equation of motion,
$(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(k,t) = e^t \int \frac{d^3p ...
1
vote
0answers
52 views
WKB expression for Dirac equation?
given the one dimensional Schroedinger equation
$$ - \frac{\hbar ^{2}}{2m} \frac{d^{2}}{dx^{2}}\Psi(x)+ V(x) \Psi(x) =E_{n}\Psi (x) $$
the WKB method for the energies is $$ (n+1)2\pi \hbar ...
1
vote
0answers
159 views
Matrix manipulation for Dirac matrices
From the Dirac equation in gamma matrices, we know that $$\gamma^i=\begin{pmatrix}
0 & \sigma^i \\
-\sigma^i & 0
\end{pmatrix}$$ and $$\gamma^0=\begin{pmatrix}
I & 0 \\
0 & -I
...
1
vote
0answers
62 views
Translate a two dimensional classical Dirac theory to a (1+1)-dim quantum theory
Suppose I have a two dimensional classical Dirac Hamiltonian with $\Psi=(\psi_1,\psi_2)^T$:
$$
H=\int \mathrm{d}x \mathrm{d}y \Psi^\dagger(\sigma^x i\partial_x+\sigma^y i\partial_y+m\sigma^z)\Psi.
$$
...
