1. Formal Second Quantization Shrödinger Equation for a Many Particle System F&W – Chapter 1 Expand multiparticle wave function in terms of a complete set of time independent single particle states where E k is a set of quantum numbers identifying the state. Enumerate the states. Then

2. Formal Second Quantization (continued) Put in Schrödinger equation and multiply by a specific set of adjoint states and integrate over all space. Most of the integrals are just normalization integrals and integrate to δ E i E’ i. The ones that don’t are the ones with the variable in the kinetic energy or potential energy term. The wave functions are assumed to be time independent; therefore only the C’s have a time derivative.

3. Formal Second Quantization (continued) Integrals are just numbers. Therefore these represent an infinite set of coupled first order differential equations for the C’s.

4. Formal Second Quantization (continued) Add particle statistics + = bosons - = fermions Now, go on to make a compact notation for bosons and fermions separately.

5. Bosons Because of bose statistics Normalization: How many times isrepresented in the sum? Permutations: Most n s are 0, but 0!=1

6. Bosons (continued) Define reduced coefficient:

7. Bosons (continued) Define new time independent wave function: Properties:

8. Boson Kinetic Energy Occupation number of E k goes down by 1; W goes up by 1. Because Letting E=i and W=j.

9. Boson Potential Energy Using the same notation for the potential:

10. Putting it all together

11. Multiply through by factorial term

12. Now go to Second Quantization Define: Orthogonal: Complete: Define set of creation and annihilation operators such that Occupation number space

13. Now Back to Schödinger’s Equation Example of one term: Relabel indices:

14. Now Back to Schödinger’s Equation Complex numbers Creation and annihilation operators in abstract number space f’s are link between quantized wave function and second quantized abstract number space.

15. End of Boson Discussion All of the statistics and operator properties are contained in the b’s End of multiparticle boson discussion: This quarter we will be looking at ground state (T=0) properties. At T=0, bosons form a condensate that has to be handled carefully. (See chapter 6 of F&W.) We will leave this to the solid state course. From here on out, we will be discussing Fermion mutiparticle states.