1. P460 - many particles1 Many Particle Systems can write down the Schrodinger Equation for a many particle system with x i being the coordinate of particle i (r if 3D) the Hamiltonian has kinetic and potential energy if only two particles and V just depends on separation then can treat as “one” particle and use reduced mass (ala classical mech. or H atom) in QM, H does not depend on the labeling. And so if any i -> j and j-> i, you get the same observables or state this as (for 2 particles) H(1,2)=H(2,1)

2. P460 - many particles2 Exchange Operator for now keep using 2 particle as example. Use 1,2 for both space coordinates and quantum states (like spin) can formally define the exchange operator as the eigenvalues of H do not depend on 1,2, implies that P 12 is a constant on motion can then define symmetric and antisymmetric states. If start out in an eigenstate, then stays in it at future times N = normalization. so for 2 and 3 particle systems

3. P460 - many particles3 Schrod. Eq. For 2 Particles have kinetic energy term for both electrons (1+2) let V 12 be 0 for now (can still make sym/antisym in any case) easy to show then that one can then separate variables and the wavefunction is: where these are (identical) single particle wavefunctions (i.e. Hydrogen, infinite box) define format. 1 (2) is particle 1’s (2’s) position and  are the quantum numbers for that eigenfunction

4. P460 - many particles4 Identical Particles Particles are represented by wave packets. If packet A has mass =.511 MeV, spin=1/2, charge= -1, then it is an electron any wave packet with this feature is indistinguishable can’t really tell the “blue” from the “magenta” packet after they overlap an ensemble (1 or more) of spin ½,3/2...particles (Fermions) have antisymmetric wavefunctions while spin 0,1,2...particles (Bosons) are described by symmetric A T0 t1>t0 t2>t1

5. P460 - many particles5 Identical Particles Create wave function for 2 particles the 2 ways of making the wavefunction are degenerate--they have the same energy--and can use any linear combination of the wavefunctions Want to have a wavefunction whose probability (that is all measured quantities) is the same if 1 and 2 are “flipped” These are NOT the same. Instead use linear combinations (as degenerate). Have a symmetric and an antisymmetric combination

6. P460 - many particles6 2 Identical Particles in a Box Create wave function for 2 particles in a box the antisymmetric term = 0 if either both particles are in the same quantum state (Pauli exclusion) OR if x1=x2 suppression of ANTI when 2 particles are close to each other. Enhancement of SYM when two particles are close to each other this gives different values for the average separation and so different values for the added term in the energy that depend on particle separation (like e-e repulsion)….or different energy levels for the ANTI and SYM spatial wave functions (the degeneracy is broken)

7. P460 - many particles7 Particles in a Box and Spin Have spatial wave function for 2 particles in a box which are either symmetric or antsymmetric there is also the spin. assume s= ½ as Fermions need totally antisymmetric: spatial ASymmetric + spin Symmetric (S=1) spatial symmetric + spin Antisymmetric (S=0) if Boson, need totally symmetric and so symmetric*symmetric or antisymmetric*antisymmetric S=1 S=0

8. P460 - many particles8 Multiparticle eigenstates Need an antisymmetric wave function (1,2,3 are positions; i,j,k…are quantum states). Can make using determinant: while it is properly antisymmetric for any 1 j. For atoms, practically only need to worry about valence effects. Solids, different terms lead to energy bands

9. P460 - many particles9 Density of States any system determine density of states D=dN/dE can do for a gas of uninteracting but overlapping identical particles density of states the same for Bosons or Fermions but how they are filled (the probability) and so average energy, etc will be different (quantum statistics – do in 461) for Fermions (i.e. electrons), Pauli exclusion holds and so particles fill up lower states at T=0 fill up states up to Fermi Energy E F. Fermi energy depends on density as gives total number of particles available for filling up states

10. P460 - many particles10 Density of States II # of available states (“nodes”) for any wavelength wavelength --> momentum --> energy “standing wave” counting often holds:often called “gas” but can be solid/liquid. Solve Scrd. Eq. In 1D go to 3D. n i >0 and look at 1/8 of sphere 0 L i.e. 2s+1

11. P460 - many particles11 Density of States III convert to momentum convert to energy depends on kinematics relativistic non-relativistic

12. P460 - many particles12 Density of States IV can find Fermi energy (assume T=0). Find number of states (n) Nonrelativistic compare to lowest energy of particle in a box of size a with  2a and this gives the condition on overlapping (particle indistinguishability). two identical particles in the same quantum state have