Transcript Helium - Northern Illinois Center for Accelerator and
Multi-Electron Atoms-Helium
• He + - same as H but with Z=2 • He - 2 electrons. No exact solution of S.E. but can use H wave functions and energy levels as starting point • nucleus screened and so Z(effective) is < 2 • “screening” is ~same as e-e repulsion (for He, we’ll look at e-e repulsion. For higher Z, we’ll call it screening) • electrons are identical particles. Will therefor obey Pauli exclusion rule (can’t have the same quantum numbers). This turns out to be due to the symmetry of the total wave function P460 - Helium 1
Schrod. Eq. For He
• have kinetic energy term for both electrons (1+2)
V T
2 2
m
1 2
V
(
r
1
T
)
V
2 2
m
(
r
2 ) 2 2
T
V
12 (
V T
r
1
T
r
2 )
E T
T
• let V12 (the e-e interaction) be 0 for now • easy to show then that one can then separate variables and the wavefunction is:
T
(
r
1 ,
r
2 ) (
r
1 )
E T
E
1
E
2 (
r
1 ) • where these are (identical) single particle wavefunctions (~that from Hydrogen) • define format. 1 (2) is particle 1’s (2’s) position and a,b are the quantum numbers for that eigenfunction b ( 1 )
or
b ( 2 ) a ( 1 )
or
a ( 2 )
or
P460 - Helium 2
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 A • any wave packet with this feature is indistinguishable T0 t1>t0 t2>t1 • can’t really tell the “blue” from the “magenta” packet after they overlap P460 - Helium 3
Identical Particles
• Create wave function for 2 particles
T
a ( 1 ) b ( 2 )
T
b ( 1 ) a ( 2 ) • the 2 ways of making the wavefunction are
E
degenerate--they have the same energy--and can
E
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 |
T
| 2 * a ( 1 ) * b ( 2 ) a ( 1 ) b ( 2 ) 1 2 * a ( 2 ) * b ( 1 ) a ( 2 ) b ( 1 ) • These are NOT the same. Instead use linear combinations (as degenerate). Have a symmetric and an antisymmetric combination
S
A
1 [ a 2 1 2 [ a ( 1 ) ( 1 ) b b ( 2 ) ( 2 ) a a ( 2 ) ( 2 ) b b ( 1 )] ( 1 )] ( 1
S A
( 1 2 ) 2 )
S
A
| | 2
unchanged
P460 - Helium 4
2 Identical Particles in a Box
• Create wave function for 2 particles in a box cos
n
x a
sin
n
x a quantum states
: a
n
1 , b
n
2
S
A
A
[cos
A
[cos
x
1
a
x
1
a
sin sin 2
x
2
a
2
x
2
a
cos
x
2
a
cos
x
2
a
sin sin 2
x
1
a
] 2
x
1
a
] • the antisymmetric term = 0 if either both particles are in the same quantum state 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 <|x2-x1|> and so different values for the added term in the energy….or different energy levels for the ANTI and SYM wave functions (the degeneracy is broken) P460 - Helium 5
Apply symmetry to He
• The total wave function must be antisymmetric • but have both space and spin components and so 2 choices:
He
space
spin
space
OR
symmetric
space
either
spin
antisym
spin
antisym
sym
• have 2 spin 1/2 particles. The total S is 0 or 1
spin
1 2 1 2 ( 1 2 1 2 1 2 ( 1 2 1 2 ) 1 2 )
S m s m s
1
m s
2 1 1 1 0 1 2 1 2 1 2 1 2 1 0 1 0 1 2 1 2 1 2 1 2 • S=1 is spin-symmetric S=0 is spin-antisymmetric P460 - Helium 6
He spatial wave function
• There are symmetric and antisymmetric spatial wavefunctions which go with the anti and sym spin functions. Note a,b are the spatial quantum numbers n,l,m but not spin
space
(
sym
)
space
(
asym
) 1 2 1 2 (
a
( 1 )
b
( 2 )
a
( 2 )
b
( 1 )) (
a
( 1 )
b
( 2 )
a
( 2 )
b
( 1 )) • when the two electrons are close to each other, the antisymmetric state is suppressed (goes to 0 if exactly the same point). Likewise the symmetric state is enhanced • --> “Exchange Force” S=1 spin state has the electrons (on average) further apart (as antisymmetric space). So smaller repulsive potential and so lower energy • note if a=b, same space state, must have S=1 (“prove” Pauli exclusion) P460 - Helium 7
He Energy Levels
• V terms in Schrod. Eq.:
V
2
e
2 4 0
r
1 2
e
2 4 0
r
2 4 0 | 2
e
r
1
r
2 | • Oth approximation. Ignore e-e term.
E T
E
1
E
2 4 13 .
6
eV n
1 2
n n
1 1
n
2 1 ( 2 ), 1
n
2 2 ( 1 ) 109 68 4 13 .
6
eV eV eV n
2 2
n
1
n
2 2 27
eV
• First approximation: look at expectation value of e-e term which will depend on the quantum states (I,j) of the 2 electrons and if S=0 or 1
E ij
E ij
V ij with V ij
4 |
e
r
1 2
r
2 | P460 - Helium 8
He Energy Levels
• For n1=n2=1 ground state, Spatial state is symmetric and S=0. The
-27
He Energy Levels
N1=2,n2=2 L1=1,L2=0 S 0 1 0 1 -70 E N1=1,n2=2 L1=0,L2=0 -110 n1=1,n2=1 S=0 P460 - Helium 10