THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 210 Assignment # 8:
 
MATRIX MADNESS!
 
Tue. 28 Oct. 2008 - finish by Tue. 4 Nov.

Since the course descriptions headlines MatLab, let's do something truly "computational" with it.

As usual, create your ~/HW/08/ directory to store your work in.

1. MATLAB WARMUP: Remember the Fibonacci numbers from Assignment 7? In a file fibmat.m, write a MatLab function to do the same thing. While you're at it, plot the resulting $F_i$ as a function of $i$ so it will be easy to check your work. Store your plot in ~/HW/08/fib.pdf (using ImageMagick's convert if necessary).

2. PAULI MATRICES: The most important matrices in Physics (so say I) are the Pauli spin matrices, described accurately in the WikipediA¹ as "a set of $2\times2$ complex Hermitian and unitary matrices . . . "

   $$\sigma_1 = \left[\begin{matrix}0 & 1 \cr 1 & 0 \end{matrix}\right . . . . . . \right]; \; \sigma_3 = \left[\begin{matrix}1 & 0 \cr 0 & -1 \end{matrix}\right]$$ (1)

   
   
   which can represent (among other things) the three components ( $\sigma_x \equiv \sigma_1$ , $\sigma_y \equiv \sigma_2$ and $\sigma_z \equiv \sigma_3$ ) of the vector spin operator $\Vec{\sigma}$ for a spin-${1\over2}$ particle.² Well, MatLab claims to be a "Matrix Laboratory", so it should be an ideal platform for verifying the essential properties of the Pauli matrices.³ Do so, for the list of properties listed on http://en.wikipedia.org/wiki/Pauli_matrices down to the beginning of the subject heading labelled "$SU(2)$ ". Make sure you understand the meaning of all these properties thoroughly.⁴

   In this notation, the spin state of a spin-${1\over2}$ particle is represented by a 2-component column vector, like

   $$\vert \! \uparrow \rangle = \left[ 1 \atop 0 \right] \qquad \hbox{\rm and} \qquad \vert \! \downarrow \rangle = \left[ 0 \atop 1 \right]$$ (2)

   
   
   for "spin up" and "spin down" (along the $\Hat{z}$ axis) respectively. Verify that operating on these column vectors from the left with the Pauli matrix $\sigma_z$ yields $+\vert \! \uparrow \rangle$ and $-\vert \! \downarrow \rangle$ , respectively.⁵

   Construct a column vector $\vert \!\! \rightarrow \rangle$ with the property that $\sigma_x \vert \!\! \rightarrow \rangle = +\vert \!\! \rightarrow \rangle$ (so that $\vert \!\! \rightarrow \rangle$ represents a spin-${1\over2}$ particle with its spin in the $+\Hat{x}$ direction).

   Similarly, construct a column vector $\vert \otimes \rangle$ with the property that $\sigma_y \vert \otimes \rangle = +\vert \otimes \rangle$ (so that $\vert \otimes \rangle$ represents a spin-${1\over2}$ particle with its spin in the $+\Hat{y}$ direction).

3. TWO SPIN-${1\over2}$ PARTICLES: Suppose you have two spin-${1\over2}$ particles, such as a proton ($p$ ) and an electron ($e$ ), whose magnetic moments $\Vec{\mu}_p = \mu_p \, \Vec{\sigma}_p$ and $\Vec{\mu}_e = -\mu_e \, \Vec{\sigma}_e$ interact with an external magnetic field $\Vec{B}$ , each contributing its Zeeman energy $E_Z = -\Vec{\mu}\cdot\Vec{B}$ . Then the Zeeman hamiltonian operator is

   $${\cal H}_Z = - \mu_p \, \Vec{\sigma}_p \cdot \Vec{B} + \mu_e \, \Vec{\sigma}_e \cdot \Vec{B} .$$ (3)

   
   
   Again picking the $\Hat{z}$ direction as the quantization axis, we have four possible fully-specified quantum states:

   $$\vert \! \Uparrow \uparrow \rangle = \left[ {1 \atop 0} \atop {0 . . . . . . \! \Downarrow \downarrow \rangle = \left[ {0 \atop 0} \atop {0 \atop 1} \right]$$ (4)

   
   where the $\Updownarrow$ and $\updownarrow$ symbols designate "spin up/down" (along the $\Hat{z}$ axis) for the electron and the proton, respectively.

   In this basis, verify that the $4\times4$ matrix representations of the electron and proton spin operators are
   

   $$\sigma_{e_1} = \left[\begin{matrix} 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 \cr 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \end{matrix}\right]; \sigma_{p_1} = \left[\begin{matrix} 0 & 1 & 0 & 0 \cr 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr 0 & 0 & 1 & 0 \cr \end{matrix}\right]$$

   
   
   

   $$\sigma_{e_2} = \left[\begin{matrix} 0 & 0 & -i & 0 \cr 0 & 0 & 0 & -i \cr i & 0 & 0 & 0 \cr 0 & i & 0 & 0 \end{matrix}\right]; \sigma_{p_2} = \left[\begin{matrix} 0 & -i & 0 & 0 \cr i & 0 & 0 & 0 \cr 0 & 0 & 0 & -i \cr 0 & 0 & i & 0 \cr \end{matrix}\right]$$ (5)

   
   
   

   $$\sigma_{e_3} = \left[\begin{matrix} 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \cr 0 & 0 & -1 & 0 \cr 0 & 0 & 0 & -1 \cr \end{matrix}\right]; \sigma_{p_3} = \left[\begin{matrix} 1 & 0 & 0 & 0 \cr 0 & -1 & 0 & 0 \cr 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & -1 \cr \end{matrix}\right]$$

   
   

   Given this information, write down the matrix representation of the full Zeeman hamiltonian for these two spins in an arbitrary magnetic field $\Vec{B} = B_x \Hat{x} + B_y \Hat{y} + B_z \Hat{z}$ . Express your result in terms of $\mu_p$ , $\mu_e$ and the three components of $\Vec{B}$ .

4. THE CONTACT INTERACTION: Suppose your two spin-${1\over2}$ particles (e.g. the proton and the electron in a hydrogen atom) interact in a way that depends only on the scalar product of their spin vectors,⁶

   $${\cal H}_{\rm hf} = A \; \Vec{\sigma}_p \cdot \Vec{\sigma}_e \; { . . . . . . ma_{e_1} + \sigma_{p_2} \sigma_{e_2} + \sigma_{p_3} \sigma_{e_3} \right) } \; ,$$ (6)

   
   
   where ${\cal H}_{\rm hf}$ is the Heisenberg hamiltonian operator and $A$ is the strength of the interaction, in energy units. For simplicity, set $A=1$ (i.e. measure all energies as multiples of $A$ ) in this part.

   Express the Heisenberg spin hamiltonian (6) as a matrix in the 4-state basis (4) defined above, and show that it is not diagonal. Using MatLab, diagonalize it and describe the new basis in which it is diagonal.⁷

5. BREIT-RABI DIAGRAM:   [EXTRA CREDIT]   We are now ready to solve the general problem of the spin hamiltonian (which governs everything the spins do!) of a hydrogen atom in an $s$ state with orbital angular momentum $\ell = 0$ .⁸ The Breit-Rabi hamiltonian is
   

   $${\cal H}_{\rm BR} = {\cal H}_{\rm hf} + {\cal H}_Z \cr\cr$$ (7)

   
   

   Express this hamiltonian in matrix form for the 4-state basis (4) and (using MatLab) diagonalize it for some particular choice of applied magnetic field, let's say $\Vec{B} = (0.1\hbox{~T}) \Hat{z}$ . Once you have accomplished this, you can repeat the diagonalization for a succession of different values of $\vert\Vec{B}\vert = B_z$ and plot the four energy eigenvalues as a function of field to get the famous Breit-Rabi diagram for hydrogen:

   Figure: : Breit-Rabi diagram showing the energy levels of a system of two spin-1/2 particles of opposite sign and different magnetic moments (e.g. the hydrogen atom) as functions of the reduced field $x \equiv B/B_0$ where $B_0$ (504.4 Oe for H in vacuum) is a characteristic hyperfine field. For the purpose of illustration, unphysical values of moments and coupling constants have been used.

   

The actual hyperfine frequency $\nu_0 \equiv \omega_0/2\pi \equiv A/h$ (where $h$ is Planck's constant) has the value 1.42040575 GHz for hydrogen in vacuum. In consistent units, $\mu_e/h = - 28.024953$  GHz/T and $\mu_p/h = 0.042577482$  GHz/T.

In zero field the three triplet ($J=1$ ) eigenstates $\vert 1\rangle$ , $\vert 2\rangle$ and $\vert 3\rangle$ are degenerate and the singlet ($J=0$ ) ground state $\vert 4\rangle$ is $\hbar\omega_0$ lower in energy.

At high reduced field ( $x \to \infty$ ) the eigenstates are
$\vert 1\rangle \to \vert\!\Uparrow\uparrow\rangle$ , $\vert 2\rangle \to \vert\!\Uparrow\downarrow\rangle$ , $\vert 3\rangle \to \vert\!\Downarrow\downarrow\rangle$ and $\vert 4\rangle \to \vert\!\Downarrow\uparrow\rangle$ .
That is, the original basis!