## PHY456H1F: Quantum Mechanics II. Lecture 16 (Taught by Prof J.E. Sipe). Hydrogen atom with spin, and two spin systems.

Posted by peeterjoot on November 2, 2011

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# The hydrogen atom with spin.

READING: what chapter of [1] ?

For a spinless hydrogen atom, the Hamiltonian was

where we have independent Hamiltonian’s for the motion of the center of mass and the relative motion of the electron to the proton.

The basis kets for these could be designated and respectively.

Now we want to augment this, treating

where is the Hamiltonian for the spin of the electron. We are neglecting the spin of the proton, but that could also be included (this turns out to be a lesser effect).

We’ll introduce a Hamiltonian including the dynamics of the relative motion and the electron spin

Covering the Hilbert space for this system we’ll use basis kets

Here should be understood to really mean . Our full Hamiltonian, after introducing a magnetic pertubation is

where

and

For a uniform magnetic field

We also have higher order terms (higher order multipoles) and relativistic corrections (like spin orbit coupling [2]).

# Two spins.

READING: section 28 of [1].

**Example**: Consider two electrons, 1 in each of 2 quantum dots.

where and are both spin Hamiltonian’s for respective 2D Hilbert spaces. Our complete Hilbert space is thus a 4D space.

We’ll write

Can introduce

Here we “promote” each of the individual spin operators to spin operators in the complete Hilbert space.

We write

Write

for the full spin angular momentum operator. The component of this operator is

So, we find that are all eigenkets of . These will also all be eigenkets of since we have

Are all the product kets also eigenkets of ? Calculate

For the mixed terms, we have

So

Since we have set our spin direction in the z direction with

We have

And are able to arrive at the action of on our mixed composite state

For the action on the state we have

and on the state we have

All of this can be assembled into a tidier matrix form

where the matrix is taken with respect to the (ordered) basis

However,

It should be possible to find eigenkets of and

An orthonormal set of eigenkets of and is found to be

latex s = 1$ and } \\ \frac{1}{{\sqrt{2}}} \left( {\left\lvert {+-} \right\rangle} + {\left\lvert {-+} \right\rangle} \right) & \mbox{ and } \\ {\left\lvert {–} \right\rangle} & \mbox{ and } \\ \frac{1}{{\sqrt{2}}} \left( {\left\lvert {+-} \right\rangle} – {\left\lvert {-+} \right\rangle} \right) & \mbox{ and }\end{array}\end{aligned} \hspace{\stretch{1}}(3.37)$

The first three kets here can be grouped into a triplet in a 3D Hilbert space, whereas the last treated as a singlet in a 1D Hilbert space.

Form a grouping

Can write

where the and here refer to the spin index .

## Other examples

Consider, perhaps, the state of the hydrogen atom

Consider the Hilbert space spanned by , a dimensional space. How to find the eigenkets of and ?

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

[2] Wikipedia. Spin.orbit interaction — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 2-November-2011]. http://en.wikipedia.org/w/index.php?title=Spin\%E2\%80\%93orbit_interaction&oldid=451606718.

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