Wednesday, March 10, 2010

To commute or not to commute

This weeks reading from Advanced Solid State Physics by Philip Phillips is Chapter 3, Second Quantization. Some key ideas:

Quasi-particles in quantum many-body theory are either fermions or bosons. The difference is seen in whether their creation and annihilation operators obey anti-commutation or commutation relations.

The second quantisation formalism provides a convenient and powerful book-keeping device to keep track of the fermionic (or bosonic) character of many-particle states.

A key result to become familiar with is writing a many-body Hamiltonian operator in second-quantised form, as in equation (3.33).
This is the starting point of Chapter 4, The Hartree-Fock approximation.

In the next chapter we will see how the anti-symmetric (fermionic) character of many-electron quantum states has an important physical consequence: the exchange interaction which leads to an energy splitting of singlet and triplet excited states. [A clear discussion of this is in Chapter 18, The Helium Atom, in Quantum Physics by Stephen Gasciorowicz.]


  1. I missed the group again. Probably this will keep up for a while until things settle down.

    I was not terribly impressed that Phillips did not go further into how you take an expectation value of a second quantized expression, nor did he put in any discussion of the difference between different relevant vacua.

    It has not impressed me at _all_ that most texts on many-body theory do not do this justice. I certainly think it is _extremely_ important. What good is knowing how to write an operator if you can't take its expectation? Understanding the identities that hold for the expectation (or higher moments) is critical for understanding how statistical models can be created.

    This is OBVIOUSLY important - for example, didn't Hubbard simulate his model instead of solving it, by making the one-body potential a random variable? How can anyone understand the cleverness of this idea unless they understand how to manipulate expectation values.

    It's possible that he mentioned it and I missed it, but for this to be true he must not have given it much discussion.

    I was even LESS impressed when I saw that one of the exercises in the back asks the reader to actually take an expectation.

    Very bad form, if you ask me.

  2. Oh, and BTW, I _HIGHLY_ recommend Lipkin's "Quantum Mechanics: New Approaches to Selected Topics" for a lucid and very nicely pedagogical discussion of second quantization. The book is excellent generally, and reads more like a novel than a textbook (even though it's all physics). Curl up in bed with one! Its a Dover edition, too, so its something less than $600/copy.