Are these matrix definitions correct for the Dirac equation?

[Orion1]

Hydrogen normalized position wavefunctions in spherical coordinates:
$$\Psi_{n \ell m}\left(r,\theta,\phi\right) = \sqrt{{\left( \frac{2}{n r_1} \right)}^3 \frac{\left(n - \ell - 1\right)!}{2n\left[\left(n + \ell\right)!\right]}} e^{-\frac{r}{n r_1}} \left({2r \over {n r_1}}\right)^{\ell} L_{n - \ell - 1}^{2 \ell + 1}\left(\rho\right) \cdot Y_{\ell}^{m}\left(\theta, \phi\right)$$

Time independent Schrödinger equation:
$$i \hbar \frac{\partial \Psi\left(\mathbf{r},\,t\right)}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi\left(\mathbf{r},\,t\right) + V\left(\mathbf{r}\right)\Psi\left(\mathbf{r},\,t\right)<br /> \end{equation*}$$

Three dimensional Laplace operator:
$$\nabla^2 \Psi\left(\mathbf{r},\,t\right) = \frac{1}{r^2 \sin \theta} \left[\sin \theta \frac{d}{dr} \left(r^2 \frac{d\Psi}{dr}\right) + \frac{d}{d \theta}\left(\sin \theta \frac{d \Psi}{d\theta}\right) + \frac{1}{\sin \theta} \frac{d^2 \Psi}{d\phi^2}\right]$$

Differential time independent Hydrogen Schrödinger equation:
$$E\left(r\right) \Psi \left(r,\theta,\phi\right) = - \frac{\hbar^2}{2 \mu} \frac{1}{r^2 \sin \theta} \left[\sin \theta \frac{d}{dr} \left(r^2 \frac{d\Psi}{dr}\right) + \frac{d}{d \theta}\left(\sin \theta \frac{d \Psi}{d\theta}\right) + \frac{1}{\sin \theta} \frac{d^2 \Psi}{d\phi^2}\right] + U\left(r\right)\Psi \left(r,\theta,\phi\right)$$

Relativistic Dirac equation:
$$i \hbar \frac{\partial \Psi\left(\mathbf{r},t\right)}{\partial t} = \left(\beta mc^2 + \sum_{k = 1}^3 \alpha_k p_k \, c\right) \Psi \left(\mathbf{r},t\right) + V\left(\mathbf{r}\right)\Psi\left(\mathbf{r},\,t\right)<br /> \end{equation*}$$

If the Schrödinger equation Laplace operator is describing a plane wave in spherical coordinates, what type of wave and coordinates does the relativistic Dirac equation describe?

What is the solution to the relativistic Hydrogen Dirac equation that is analogous to the differential time independent Hydrogen Schrödinger equation?
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Reference:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html"
http://en.wikipedia.org/wiki/Hydrogen_atom#Mathematical_summary_of_eigenstates_of_hydrogen_atom"
http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation"
http://en.wikipedia.org/wiki/Dirac_equation"

 

[Bill_K]

These equations are general, there's no implication that a plane wave is involved. The Laplacian in the Schrödinger equation is momentum squared. To make the Dirac Equation look analogous replace p_k by ∇_k and specialize to spherical coordinates.

For the Dirac Equation for a Coulomb field, a separation-of-variables solution can be found. The angular part is a Y_lm with half-integer m.

 

[dextercioby]

So you seek answers to the 2 questions ? Well, the first doesn't make too much sense to me, because the Laplace operator can be expressed roughly in 12 different coordinate systems (check out Morse & Feshbach's 2 volumes from the 1950s), in all of which the variables can be separated, so that the <spherical ones> are by no means special, except probably for the H-atom.

OTOH, what makes you think Dirac's equation(s) describe waves ?

For your second question, if you haven't found the answer on Wikipedia, then it must not exist. :) Just kidding, IIRC Messiah's second volume contains a useful discussion, one (I don't remember which) of Greiner's volumes also and most certainly other sources as well (Input is welcome).

 

[alxm]

For your second question, if you haven't found the answer on Wikipedia, then it must not exist. :) Just kidding, IIRC Messiah's second volume contains a useful discussion, one (I don't remember which) of Greiner's volumes also and most certainly other sources as well (Input is welcome).

Bethe and Salpeter has a fairly detailed treatment. So do some quantum-chemistry textbooks (provided they deal with relativistic theory at all), one is Piela's recent "Ideas of Quantum Chemistry".


I'm not sure about the premise of the question here. There really isn't any 'analogous' solution for the Hydrogen Dirac equation, because the Dirac equation isn't an analogue of the Schrödinger equation. It just looks similar.

 

[ytuab]

Similar point is that the boundary conditions (radial) of Schrödinger and Dirac equations are from zero to infinity.
(Because both of them use the exponential function, which becomes zero at infinity.)

So the de Broglie's relations are applied to match this boundary condition.

Different point is that Dirac equation needs four component spinor.
Usual Dirac equation needs all four components to express one state.
But Dirac euqation for hydrogen contains two states (l = j + 1/2 and l = j - 1/2) in one four component spinor.
So one is real and another is imaginary.
(Of course, only one of them can not get the solution.)

Most different thing is that the frequencies (= differentiation by time t) are completely different in them.
Because Dirac equation includes great rest mass energy of

$$mc^{2}$$

So the energy (=frequency) is much greater than that of Schrödinger equation, though the relativistic effect of hydrogen atom is very small.
( Of course, the momentum parts of them are similar. But which frequency is real ? )

 

[Orion1]

I tried to extrapolate the following equations from references.

Dirac equation:
$$i \hbar \frac{\partial \Psi\left(\mathbf{r},t\right)}{\partial t} = \left(\beta mc^2 + \sum_{k = 1}^3 \alpha_k p_k \, c\right) \Psi \left(\mathbf{r},t\right) + V\left(\mathbf{r}\right) \Psi\left(\mathbf{r},\,t\right)$$

Pauli matrices:
$$\alpha_k = \left(\begin{array}{cc} 0 & \sigma_k \\ \sigma_k & 0 \\ \end{array}\right)_k$$

$$\beta = \left(\begin{array}{cc} 0 & I_{2} \\ I_{2} & 0 \\ \end{array}\right)$$

Dirac momentum:
$$p_k = -i \hbar \nabla_k$$

Relativistic Hydrogen Dirac equation:
$$E\left(r\right) \Psi \left(r,\theta,\phi\right) = \left(\left(\begin{array}{cc} 0 & I_{2} \\ I_{2} & 0 \\ \end{array}\right) m_{e} c^2 - i \hbar c \sum_{k = 1}^3 \left(\begin{array}{cc} 0 & \sigma_k \\ \sigma_k & 0 \\ \end{array}\right)_k \nabla_k\right) \Psi \left(r,\theta,\phi\right) + U\left(r\right) \Psi\left(r,\theta,\phi\right)$$

Is this equation symbolically correct at this point?

According to reference 2, the matrix definitions are:
$$\alpha_1 = \left(\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array}\right)$$

$$\alpha_2 = \left(\begin{array}{cccc} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & -i & 0 & 0 \\ i & 0 & 0 & 0 \\ \end{array}\right)$$

$$\alpha_3 = \left(\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ \end{array}\right)$$

$$\beta = \left(\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array}\right)$$
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Reference:
http://en.wikipedia.org/wiki/Dirac_equation"
http://www.fermentmagazine.org/seminar/dirac.pdf"
http://www7b.biglobe.ne.jp/~kcy05t/dirachy.html"