💼 Case: lets try to find the probability density using the Dirac equation

• Starting from the Dirac equation

  $$ i\hbar \frac{\partial \psi}{\partial t} =-i\hbar c\vec \alpha\cdot \vec \nabla \psi + \beta mc^2 \psi
  $$

• Multiplying from the left by $\psi^\dag$ gives

  $$ i\hbar \psi ^ \dag\frac{\partial \psi}{\partial t} =-i\hbar c\psi ^ \dag\, \vec \alpha\cdot \vec \nabla \psi + mc^2 \psi ^\dag \beta\psi
  $$

  🗒️ Note: here we used $\psi^\dag$ rather than $\psi ^*$ because our wave-function is a vector not a scalar

• We can build the adjoint matrix and multiplying by $\psi$ from the right gives

  $$ -i\hbar \frac{\partial \psi^ \dag}{\partial t}\psi =i\hbar c \vec \nabla \cdot ( \psi ^ \dag\vec \alpha) \psi + mc^2 \psi ^\dag \beta\psi
  $$

  where $\vec \alpha$ and $\beta$ are Hermitian and thus are not affected

• Subtracting our previous and current expression gives

  $$ i\hbar \frac{\partial}{\partial t} ( \psi ^ \dag \psi ) = -i\hbar c \vec \nabla \cdot (\psi^\dag \vec \alpha \psi) $$

  which we recognise as

  $$ \frac{\partial \rho}{\partial t} +\vec \nabla \cdot \vec j =0 $$

  So the probability density and the current are

  $$ \rho = \psi ^ \dag \psi \qquad \vec j =\psi ^ \dag \psi $$

  • 🗒️ Note: $\rho$ can be interpreted as a probability density again and we can use to normalise the wave function by integrating over all space

Angular momentum and spin

⚽ Goal: lets try and determine wether the orbital angular momentum is conserved

• The orbital angular momentum operator is defined as

  $$ \hat {\vec L } =\hat{\vec r} \times \hat{\vec p} \qquad \Leftrightarrow \qquad \hat L_i =\sum_{jk}\epsilon_{ijk}\hat r_j \hat p_k $$

• Where our goal is to find

  $$ [\hat H, \hat{\vec L}]=[v\vec \alpha \cdot \vec p +\beta mc^2 ,\hat{\vec L}]=? $$

• Calculating it directly we get

  $$ [\hat H , \hat{\vec L}]=-i\hbar c\vec \alpha \times \hat{\vec p} $$

  💎 Conclusion: The orbital angular momentum is not conserved

• We expect the total angular momentum to be conserved so lets introduce the spin

  $$ \vec \Sigma =\begin{bmatrix} \vec \sigma & 0 \\ 0 & \vec \sigma \end{bmatrix} \qquad \Leftrightarrow \qquad \Sigma _i=\begin{bmatrix} \sigma_i & 0 \\ 0 & \sigma_i \end{bmatrix} \quad \text{for} \;i\in \{1,2,3\} $$

  where $\vec \sigma$ are the Pauli matrices

• Calculating its commutation relation we have

  $$ [\alpha_i , \Sigma _j]=2i \sum k \epsilon{ijk} \alpha_k \quad "+" \quad [\beta ,\Sigma_j]=0 \quad "\Rightarrow" \quad [\hat H,\vec \Sigma ]=2ic\,\vec \alpha \times \hat{\vec p} $$

• Hence by considering the total angular momentum, $\hat{\vec J} =\hat{\vec L}+\frac{\hbar}{2}\vec \Sigma \equiv \hat {\vec L} +\hat {\vec S}$ we write

  $$ [\hat H ,\hat {\vec L}+\hat{\vec S} ]=0 $$

  💎 Conclusion: The total angular momentum is conserved and the spin operator is $\hat {\vec S}=\frac{\hbar}{2}\vec \Sigma$

Taking a closer look at the Spin operator we have

• Commutation: $[\hat S_i,\hat S_j]=i\hbar \sum_k \epsilon_{ijk}\hat S_{k}\quad \text{for} \quad i,j,k\in \{ 1,2,3\}$
• Eigenvalues: $\hat S_i = \pm \hbar/2$ and thus describes a spin-$\frac 12$ particle

Plane wave states

⚽ Goal: find the solution to the free particle Dirac equation

• Lets use as our Ansatz the following plane wave

  $$ \psi ( \vec r,t)=Nu(\vec p) e^{i(\vec p \cdot \vec r - Et)/\hbar} =Nu(\vec p ) e^{-ip^\mu r_\mu /\hbar} $$

  where $N$ is some normalisation and $u(\vec p )$ is a four-component spinor

  We can express $u(\vec p )$ as follows

  $$ u(\vec p ) =\begin{pmatrix} \phi \\ \chi \end{pmatrix} $$

  where $\phi$ and $\chi$ are known as the upper and lower components of the wave functions