1.3. MANY PARTICLES |
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Rotationally invariant harmonic trap |
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m |
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U (~r) = |
(wr2r2 + wz2z2) |
r = px2 + y2 |
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2 |
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This leads to the solution |
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ˆ |
imf |
jnz |
(zˆ) |
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with |
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ynr mnz = jnr (r)e |
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r |
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z |
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r = |
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l |
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zˆ = |
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i l |
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q |
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r |
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q |
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z |
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mwr |
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mwz |
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9
(1.12)
(1.13)
(1.14)
jnz |
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1 |
e |
zˆ2 |
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= |
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Hnz (zˆ) |
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(1.15) |
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2 |
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p |
p |
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2nz nz!lz |
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n |
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jnr |
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r |
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2 |
) |
(1.16) |
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= lr sp(nr + jmj)! rˆ jmje |
2 Lnj rj(rˆ |
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1 |
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n ! |
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rˆ 2 |
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enr mnz |
= ~(wr (2nr + jmj+ 1) + wz(nz + |
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(1.17) |
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2 |
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Here H are the HERMITE polynomials and L the LAGUERRE polynomials.
Isotropic harmonic trap |
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mw2 |
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U (~r) = |
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r2 |
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(1.18) |
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The dimensionless solutions are |
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ynlm = jnl (~r)Ylm(~r) |
rˆ = |
r |
rl0 |
(1.19) |
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mw |
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nl |
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l 2 |
s |
Γ(n + l + |
23 ) |
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j |
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2n! |
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rˆl e |
rˆ2 |
Lnl+ 21 (rˆ |
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enlm |
= ~w(2n + l + |
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2 |
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Here Γ is the EULER gamma (generalized faculty) function.
1.3Many particles
The HAMILTONian is
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N |
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~2 |
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1 |
N |
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(~ri |
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i=1 |
2m 4 |
2 i= j=1 |
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H = å |
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i +U (~ri) + |
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å |
UI |
~r j): |
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6 |
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(1.20)
(1.21)
(1.22)
10 |
CHAPTER 1. GENERAL ASPECTS |
The simplest case is the free gas, where UI 0. If we label possible single particle states with n and if each state is occupied by nn particles than the total number of particles N is
ånn = N: |
(1.23) |
n |
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A simple ansatz for the wave function is a product of single particle wave functions:
Ψn1;n2;:::(~r1;~r2;:::;~rn) = ji1 (~r1)ji2 (~r2) jin (~rn) |
(1.24) |
en1;n2;::: = n1e1 + n2e2 + ::: |
(1.25) |
This solution satisfies the S CHRÖDINGER equation (1.5) but, in general, it fails to describe the physics of N identical particles1, because the wave function must change in a specific way under permutations P of any two identical particles. Since jyj2 is an observable which is unaffected by the permutation, this leaves two possibilities:
Py = y |
(1.26) |
1."+": Bosons
The wave function has to be symmetrized over all possible permutations which exchange particles in different quantum states. This subset of the
permutation group is denoted by p0:
yn1;n2;::: = r |
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jn1 (~r1) jnn (~rn) |
(1.27) |
1 |
N! åp0 |
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(b) |
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n |
!n2! |
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2." ": Fermions
No two single particles may be in the same state n. Therefore the sum runs over all possible permutations:
(f) |
1 |
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åp |
p |
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yn1;n2;::: = |
p |
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jn1 (~r1) jnn (~rn) |
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N! |
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Alternatively the wave function can be described by a determinant:
(f) |
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yn1;n2;::: = pN!
0jn2 |
(~r1) jn2 |
(~r2) |
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jn1 |
(~r1) |
jn1 |
(~r2) |
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.. |
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.. |
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B |
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Bj |
nn |
(~r |
1 |
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nn |
(~r |
2 |
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B |
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@ |
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1i.e. no further quantum numbers like color is present
::: |
jn1 (~rn) |
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::: |
jn2 (~rn)1 |
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. . |
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C |
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C |
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C |
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A |
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::: |
jnn (~rn) |
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(1.28)
(1.29)