Dirac Definition and 900 Threads

Hi, I'm studying direct detection techniques for dark matter and in almost all the articles I read (e.g. Gondolo, P. (1996, May 13). Phenomenological Introduction to Direct Dark Matter Detection. arXiv.org.) the authors say that in the non-relativistic limit the vector and axial currents take...

Homework Statement Show that $$\delta(k^2) \delta[(k-q_2)^2] = \delta(k^2) \delta(k^0 - \sqrt{s}/2) \frac{1}{2\sqrt{s}},$$ where ##k = (k^0, \mathbf k)## and ##s = q_2^2,## where ##q_2 = (\sqrt{s},\mathbf 0)## 2. Homework Equations I was going to use the fact that $$\delta(f(x)) = \sum...

Is the following equation mathematically rigorous? How can you tell? ## \int_{-\infty}^\infty \delta(x-a) \delta(x-b) dx=\delta(a-b)## Thanks

Hey I was reading through a text and came across: I can understand the second statement from the Pauli matrices... However I think that I don't understand the 1st statement as it is... why would the diagonal elements of an odd-operator be zero if parity is definite?

Homework Statement Homework Equations Dirac equation: $$(\gamma ^{\mu}\rho_{\mu}-mc)\psi=0$$ The Attempt at a Solution If we multiply out the Dirac equation by inserting all it's components we get: which if I've multiplied it correctly gives $$ \begin{bmatrix} \frac{E}{c}-mc\\ 0 \\...

Consider: ##\nabla^{2} V(\vec{r})= \delta(\vec{r})## By taking the Fourier transform, the differential equation dissapears. Then by transforming that expression back I find something like ##V(r) \sim \frac{1}{r}##. I seem to have lost the homogeneous solutions in this process. Where does this...

Hi I read that for Dirac equation, [ L , H ] =/ 0 , so Dirac found a operator S such that 1. [ S , H ] = - [L, H] ---> [ J, H ] = 0 where J = L + S , the total angular momentum. 2. S gives an eigenvalue of spin 1/2 ---> solutions of Dirac equation describe fermions. The total...

Homework Statement The free Dirac equation is given by ##(i\gamma ^\mu \partial _\mu -m)\psi = 0## where ##m## is the particle's mass and ##\gamma ^\mu## are the Dirac gamma matrices. Show that for the equation to be consistent with Relativity, the gamma matrices must satisfy ##[\gamma ^\mu...

Hey guys, So here's the deal. Consider the Lagrangian \mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi where \bar{\psi}=\psi^{\dagger}\gamma^{0} . I need to find the Hamiltonian density from this, using \mathcal{H}=\pi_{i}(\partial_{0}\psi_{i})-\mathcal{L} So I get the following...

Homework Statement [/B] I'm supposed to calculate the advanced propagator for the Dirac field, and I have no problem with that. Then I'm supposed to show it vanishes for spacelike separation (that is (x-y)^2<0). Homework Equations For the advanced propagator I get something like: S_A =...

Does a completely regular space imply the Dirac measure. From wikipedia we have the definition: X is a completely regular space if given any closed set F and any point x that does not belong to F, then there is a continuous function, f, from X to the real line R such that f(x) is 0 and, for...

Hi everybody, I was doing one asignment form class, I was tasked to prove that in one system, the arimetic mean of FD and BE distributions is equal to MB's distribution for undishtingable particles. After doing the numbers I found out that it actually was, but I don't know why this happens, can...

Homework Statement Consider the Dirac Hamiltonian ##\hat H = c \alpha_i \hat p_i + \beta mc^2## . The operator ##\hat J## is defined as ##\hat J_i = \hat L_i + (\hbar/2) \Sigma_i##, where ##\hat L_i = (r \times p)_i## and ##\Sigma_i = \begin{pmatrix} \sigma_i & 0 \\0 & \sigma_i...

When talking about the strength of a delta potential , the delta potential is multiplied by a parameter ie α but how does a delta potential have a strength ? It is zero everywhere and infinite at x = 0. The parameter makes no difference to zero or infinity.

Dirac had proposed that Gravitational constant would reduce with time.Why?

Hi, is the wave function that couples to the Dirac equation the same as that which couples to the Schrodinger equation? Thanks.

I've been thinking about the properties of the Dirac delta function recently, and having been trying to prove them. I'm not a pure mathematician but come from a physics background, so the following aren't rigorous to the extent of a full proof, but are they correct enough? First I aim to...

Homework Statement Having trouble understanding dirac deltas, I understand what they look like and how you can express one (i.e. from the limiting case of a gaussian) but for the life of me I can't figure out why the results of some integrals featuring dirac deltas equate to what they do...

I have been reading through Mark Srednicki's QFT book because it seems to be well regarded here at Physics Forums. He discusses the Dirac Equation very early on, and then demonstrates that squaring the Hamiltonian will, in fact, return momentum eigenstates in the form of the momentum-energy...

Homework Statement A measurement is described by the operator: |0⟩⟨1| + |1⟩⟨0| where, |0⟩ and |1⟩ represent orthonormal states. What are the possible measurement outcomes? Homework Equations [/B] Eigenvalue Equation: A|Ψ> = a|Ψ> The Attempt at a Solution Apologies for the basic...

For a research project, I have to take multiple derivatives of a Yukawa potential, e.g. ## \partial_i \partial_j ( \frac{e^{-m r}}{r} ) ## or another example is ## \partial_i \partial_j \partial_k \partial_\ell ( e^{-mr} ) ## I know that, at least in the first example above, there will be a...

Homework Statement ∫δ(x3 - 4x2- 7x +10)dx. Between ±∞. Homework EquationsThe Attempt at a Solution Well I don't really know how to attempt this. In the case where inside the delta function there is simply 2x, or 5x, I know the answer would be 1/2 or 1/5. Or for say δ(x^2-5), the answer would...

Solutions to the Klein-Gordon equation can be interpreted as spin-0 particles of mass m and charge +1, 0, -1? See here. I think same can be said for solutions to the Dirac equation in 1+1 dimensional space-time, solutions can be interpreted as spin-0 particles of mass m and charge +1, 0, -1...

I have not read any other QM books,i have little knowledge on that subject and want a books that uses mathematics in academic levels but is easy to get the grips on and also builds intuition and explains the phenomenons in a good manner.I do not want a book that emphasizes on mathematics or...

What type of "graph paper" do I need to graph an arbitrary solution, Ψ, of the Dirac equation in 3+1 dimensional spacetime? Assume the "graph paper" has the minimum dimensions required to do the job. Would this work? At each point of spacetime we need a complex plane which takes care of the...

Homework Statement Consider the Hamiltonian: $$\hat{H}=C*(\vec{B} \cdot \vec{S})$$ where $C$ is a constant and the magnetic field is given by $$\vec{B} = (0,B,0) $$ and the spin is $$\vec{S} = (\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}),$$ with$$\hat{S}_{x}...

Dirac description If I well understood a Dirac description for fermions is : ##\Psi_{D}=\Psi_{L}+\Psi_{R}## where ##\Psi_{L}## is the left-chiral spinor and ##\Psi_{R}## the right-chiral spinor. Each spinor, ##\Psi_{L} ## and ##\Psi_{R}## has 2 components cotrresponding to the particle and...

I'm trying to solve the following equation (even if I'm not sure if it's well posed) \partial_{x} \, y(x) + a(x)\, y(x) = \delta(x) with ##\quad \lim_{x \rightarrow \pm \infty}y(x) = 0## It would be a classical first order ODE If it were not for the boundary conditions and the Dirac...

Hey everybody, I'm an engineering Ph.D. so my knowledge of n-dimensional Euclidean spaces is lacking to say the least. I'm wondering what sort of approach I can take to solve this problem. ##\boldsymbol{1.}## and ##\boldsymbol{ 2. }## I am given a probability distribution for a random...

Homework Statement Evaluate the integrals in the attached image Homework EquationsThe Attempt at a Solution

The original Dirac Equation was for the electron, a particle of spin 1/2. Is there a "Generalized Dirac Equation" that has been experimentally proven to work for all fermions, not just those of spin 1/2? Thanks in advance.

Homework Statement Solve the integral ## \int_0^{3\pi} \delta (sin \theta) d\theta## Homework EquationsThe Attempt at a Solution I can rewrite ## delta (sin \theta) ## as ##\sum_{n=-\infty}^{\infty} \frac{\delta(\theta - n\pi)}{|cos (n\pi)|}=\sum_{n=-\infty}^{\infty} \delta(\theta-n\pi)## So...

In Dirac's book on relativity, he begins and ends his section on proving the stationary property of geodesics with references to "null geodesics". His last sentence is: "Thus we may use the stationary condition as the definition of a geodesic, except in the case of a null geodesic." What is a...

Is it a must to know clifford algebra in order to derive the dirac equation? I recently watch drphysics video on deriving dirac equation and he use two waves moving in opposite directions to derive it, without touching clifford algebra. If this possible, what is the intuition behind it?

Can anyone give me a really simple example on how to use the eqn above to solve it? The eqn is the modified schrodinger eqn that takes into account relativity.

In Dirac's book on GRT, top of page 17, he has this: (I'll use letters instead of Greeks) gcdgac(dva/ds) becomes (dvd/ds) I seems to me that that only works if the metric matrix is diagonal. (1) Is that correct? (2) If so, that doesn't seem to be a legitimate limitation on the property of...

Homework Statement Break integral into positive and negative, integrate, recombine and simplify and show that it reduces to a real-valued function. (See attachments) Homework Equations See attachments The Attempt at a Solution My solution is not reducing to a real-valued function. Please see...

The Dirac equation in 3+1 space-time yields spin, is this still true in 1+1d space-time? If not what do the 2 components of the spinor represent? Do we still have intrinsic spin in 1+1d space-time? Thanks for any help!

I found a paper that derives the Dirac equation in 1 + 1 dimensional space-time. It is equation 8, here, http://academic.reed.edu/physics/faculty/wheeler/documents/Classical%20Field%20Theory/Miscellaneous%20Essays/A.%202D%20Dirac%20Equation.pdf and here...

I've taken a course in QM 1, based on the Schrodinger picture and QM 2 looks to be a continuation of this picture. Looking through Wikipedia, I found the article on the Dirac picture. Is there a good undergraduate (at the level of Griffiths or Shankar) textbook on this picture of QM? Since...

Homework Statement [/B] This is an excercise that was given by my professor in a previous test: Consider the equation: $$ \displaystyle{\not} p =\gamma^\mu p_\mu= m$$ where the identity matrix has been omitted in the second member. Find its most general solution. Homework Equations The...

How fast does the computational complexity of the Dirac equation, with regards to full* solution, grow with number of particles N? can we specify the order of time t(N) for this solution in terms of t(N=1)? (I assume that number of protons, neutrons and electrons combined is N - i.e. that...

Homework Statement I am having trouble understanding this: I have a Dirac Delta function $$ \delta (t_1-t_2) $$ but I want to prove that in the frequency domain (Fourier Space), it is: $$\delta(\omega_1+\omega_2) $$ Would anyone have any ideas how to go about solving this problem? I know...

What is a Dirac eletron ? I just take this concept when reading a news in a physics page. Thank you for helping me out.

Homework Statement Consider a one-dimensional particle subject to the Hamiltonian H with wavefunction \Psi(r,t) =\sum_{n=1}^{2} a_{n}\Psi _{n}(x)e^{\frac{-iE_{n}t}{\hbar}} where H\Psi _{n}(x)=E_{n}\Psi _{n}(x) and where a_{1} = a_{2} = \frac{1}{\sqrt{2}}. Calculate the expectation value of the...

I am confused about the coupling of the Dirac equation to electromagnetism. The 4-current that is the source for Maxwell's equation that arises from the Lagrangian \begin{equation} \mathcal{L}=i\overline{\psi}\gamma^\mu(\partial_\mu+ieA_\mu)\psi-m\overline{\psi}\psi \end{equation} is...

Homework Statement This problem came when I was learning the Poisson's equation (refer to http://farside.ph.utexas.edu/teaching/em/lectures/node31.html). when it came to the step to find the Green's function G which satisfies \nabla^2 \cdot G(\textbf{r}, \textbf{r}') =...

Homework Statement Hey guys, Consider the U(1) transformations \psi'=e^{i\alpha\gamma^{5}}\psi and \bar{\psi}'=\bar{\psi}e^{i\alpha\gamma^{5}} of the Lagrangian \mathcal{L}=\bar{\psi}(i\partial_{\mu}\gamma^{\mu}-m)\psi. I am meant to find the expression for \partial_{\mu}J^{\mu}. Homework...

Homework Statement x(t) = cos(3πt) h(t) = e-2tu(t) Find y(t) = x(t) * h(t) (ie convolution) Homework Equations Y(s) = X(s)H(s) and then take inverse laplace tranform of Y(s) The Attempt at a Solution L(x(t)) = [πδ(ω - 3π) + πδ(ω + 3π)] L(h(t)) = \frac{1}{s+2} Laplace Transform inverse ...

Homework Statement Find the conserved Noether current j^\mu of the Dirac Lagrangian L = \bar{\psi} ( i \partial_\mu \gamma^\mu - m ) \psi under the transformation: \psi \rightarrow e^{i \alpha} \psi \,\,\,\,\,\,\,\,\,\, \bar{\psi} \rightarrow e^{-i \alpha} \bar{\psi} Homework Equations...