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# sec11 - XI Identical Particles a The Product Basis We have...

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XI. Identical Particles a. The Product Basis We have already dealt with multiple particles implicitly. For example, when we were doing inelastic scattering calculations, our basis states involved specifying the state of the electron and the molecule, n ; ψ . Although we did not stress it at the time, these basis states are equivalent to products : a state for the electron times a state for the molecule: n n ψ ψ = ; To verify this, we act on the right hand state with the zeroth order Hamiltonian: ( ) ( ) ( ) n n n n n H k ψ ω ψ ω ψ 2 1 2 2 1 2 2 1 0 2 ˆ ˆ + + = + + - = and we see that these are, indeed, eigenfunctions of 0 ˆ H as advertised. This is a general rule: one can conveniently build a many particle wavefunction that describes the state of particles A, B, C, D,… by considering a product of states for A, B, C, D,… individually: ... ... D C B A D C B A ψ ψ ψ ψ ψ ψ ψ ψ = Ψ where from here on out we will use capitol Greek letters to denote many particle states and lower case Greek letters when representing one-particle states. The particles A, B, C, D,… can be any set of distinct particles (e.g. A=H 2 O, B=He+, C= H 2 O+, D=He,…). Then A ψ would be a wavefunction that represents the state of the water molecule, B ψ would represent the state of the He+, etc. What do these product states mean? To look at this question, lets assume the Hamiltonian can be decomposed into a Hamiltonian that acts just on A plus one for B plus one for C …: ... ˆ ˆ ˆ ˆ ˆ + + + + = D C B A h h h h H Many Particle ψ Single Particle ψ ’s

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This Hamiltonian describes independent particles (because it lacks coupling terms AB h ˆ that depend on A and B simultaneously). Now, the eigenstates of this independent particle Hamiltonian can always be chosen to be product states , for which: ( ) ( ) Ψ = + + + + = + + + + = Ψ E e e e e h h h h H D C B A D C B A D C B A D C B A ψ ψ ψ ψ ψ ψ ψ ψ ... ... ˆ ˆ ˆ ˆ ˆ . Thus, product states describe particles that are independent of, or uncorrelated with, one another. One consequence of this fact is that a measurement on A will not affect the state of B and vice versa. For this reason, product states are also sometimes called “independent particle” states, which is appropriate since they describe the state of each particle independently . Note that just because product states happen to describe many particles, it does not follow that every many particle wavefunction can be written in this form. In fact, it is easy to build a many particle wavefunction that is not of product form. For example: D C B A D C B A c c 2 2 2 2 2 1 1 1 1 1 2 ψ ψ ψ ψ ψ ψ ψ ψ + Ψ cannot be written as a product of one particle wavefunctions. In particular, for this state, a measurement on A influences the outcome of measurements on B, C and D: if we find that A is in state 1, B,C and D must collapse to state 1, while A in state 2 implies B, C and D will collapse to state 2. This strange correlation between different particles is called entanglement and plays a very important role in many-particle quantum mechanics. Meanwhile, the state ( ) ( ) ( ) ( ) D D C C B B A A c c c c c c c
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sec11 - XI Identical Particles a The Product Basis We have...

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