quantum mechanics and the hydrogen atom
TRANSCRIPT
MNW-L2
Since we cannot say exactly where an electron is, the Bohr picture of the atom, with electrons in neat orbits, cannot be correct.
Quantum theory describes electron probability distributions:
Quantum Mechanics and the hydrogen atom
MNW-L2
Hydrogen Atom: Schrödinger Equation and Quantum Numbers
Potential energy for the hydrogen atom:
The time-independent Schrödinger equation in three dimensions is then:
Equation 39-1 goes here.
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Where does the quantisation in QM come from ?
The atomic problem is spherical so rewrite the equation in (r,θ,φ)
φθ cossinrx = φθ sinsinry = θcosrz =
Rewrite all derivatives in (r,θ,φ), gives Schrödinger equation;
Ψ=Ψ+Ψ⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
∂∂
−Ψ⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
− ErVmr
rrm
)(sin
1sinsin
122 2
2
2
22
2
φθθθ
θθhh
This is a partial differential equation, with 3 coordinates (derivatives);Use again the method of separation of variables:
( ) ( ) ( )φθφθ ,,, YrRr =Ψ
Bring r-dependence to left and angular dependence to right (divide by Ψ):
( )( )( )
( ) λφθφθθφ =−=⎥
⎦
⎤⎢⎣
⎡−+
,,21
2
22
YYO
RrVEmrdrdRr
drd
R
QM
h Separation of variables
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Where does the quantisation in QM come from ?
( )( ) λ=⎥⎦
⎤⎢⎣
⎡−+ RrVEmr
drdRr
drd
R 2
22 21
hRadial equation
Angular equation ( )( )
( )
( ) λφθ
φθφθθ
θθθ
φθφθθφ =
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂+
∂∂
∂∂−
=−,
,sin
1sinsin
1
,, 2
2
2
Y
Y
YYOQM
YYY θλθ
θθ
θφ
22
2sinsinsin +
∂∂
∂∂
=∂∂
−
Once more separation of variables: ( ) ( ) ( )φθφθ ΦΘ=,Y
Derive: 222
2sinsinsin11 m=⎟
⎠⎞
⎜⎝⎛ Θ+
∂Θ∂
∂∂
Θ=
∂Φ∂
Φ− θλ
θθ
θθ
φ(again arbitrary constant)
Simplest of the three: the azimuthal angle;
( ) ( ) 022
2=Φ+
∂Φ∂ φφφ m
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Where does the quantisation in QM come from ?
A differential equation with a boundary condition
( ) ( ) 022
2=Φ+
∂Φ∂ φφφ m and ( ) )(2 φπφ Φ=+Φ
Solutions:
( ) φφ ime=Φ
Boundary condition; ( ) ( ) ( ) φπφ φπφ imim ee =Φ==+Φ +22
12 =ime π
m is a positive or negative integerthis is a quantisation condition
General: differential equation plus a boundary condition gives a quantisation
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Where does the quantisation in QM come from ?
( ) φφ ime=ΦFirst coordinatewith integer m (positive and negative)
Second coordinate 0sin
sinsin
12
2=Θ⎟
⎟⎠
⎞⎜⎜⎝
⎛−+
∂Θ∂
∂∂
θλ
θθ
θθm
Results in ( )1+= lllλ with Kl ,2,1,0=
and llKll ,1,,1, −+−−=m
angularpart
angularmomentum
Third coordinate ( )( ) ( )1212
22 +=⎥
⎦
⎤⎢⎣
⎡−+ ll
hRrVEmr
drdRr
drd
R
Differentialequation Results in quantisation of energy
2
2
0
2
2
2
2
2
24 h
en
menZR
nZE ⎟
⎟⎠
⎞⎜⎜⎝
⎛−=−= ∞ πε
with integer n (n>0)
radialpart
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Angular wave functions
⎥⎦
⎤⎢⎣
⎡
∂∂
+∂∂
∂∂
= 2
2
22
sin1sin
sin1
φθθθ
θθiL h
with Kl ,2,1,0= and llKll ,1,,1, −+−−=m
Operators:
φ∂∂
=i
Lzh
There is a class of functions that are simultaneous eigenfunctions
),,( zyx LLLL =r
Angular momentum
( ) ( ) ( )φθφθ ,1, 22lmlm YYL hll += ( ) ( )φθφθ ,, lmlmz YmYL h=
Spherical harmonics (Bolfuncties) ( )φθ ,lmY
π41
00 =Y φθπ
ieY sin83
11 −=
θπ
cos43
10 −=Y
φθπ
ieY −− = sin
83
1,1
Vector space of solutions
( ) 1, 2 =Ω∫Ω
dYlm φθ
''''*
mmllmllm dYY δδ=Ω∫Ω
( ) ( ) ( ) ( )φθπφθπφθ ,,, lmlmopP Υ−=+−Υ=Υ l
Parity
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The radial part: finding the ground state
Find a solution for 0,0 == ml
ERRr
ZeRr
Rm
=−⎟⎠⎞
⎜⎝⎛ +−
0
22
4'2"
2 πεh
Physical intuition; no density for ∞→r
trial: ( ) arAerR /−=
aRe
aAR ar −=−= − /'
2/
2"aRe
aAR ar == −
Er
Zearam
=−⎟⎠⎞
⎜⎝⎛ −−
0
2
2
2
421
2 πεh
must hold for all values of r
04 0
22=−
πεZe
mahPrefactor for 1/r:
mZea 2
20
04 hπε
=
Solution for thelength scale paramater
Bohr radius
Solutions for the energy
2
2
0
22
2
242 h
h emeZma
E ⎟⎟⎠
⎞⎜⎜⎝
⎛−=−=
πε
Ground state in the Bohr model (n=1)
( )( ) λ=⎥⎦
⎤⎢⎣
⎡−+ RrVEmr
drdRr
drd
R 2
22 21
h
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There are four different quantum numbers needed to specify the state of an electron in an atom.
1. The principal quantum number n gives the total energy.
2. The orbital quantum number gives the angular momentum; can take on integer values from 0 to n - 1.
Hydrogen Atom: Schrödinger Equation and Quantum Numbers
ll
3. The magnetic quantum number, m , gives the direction of the electron’s angular momentum, and can take on integer values from – to + .l l
l
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Hydrogen Atom Wave Functions
The wave function of the ground state of hydrogen has the form:
The probability of finding the electron in a volume dVaround a given point is then |ψ|2 dV.
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Radial Probability Distributions
Spherical shell of thickness dr, inner radius rand outer radius r+dr.
Its volume is dV=4πr2dr
Density: |ψ|2dV = |ψ|24πr2dr
The radial probablity distribution is then:
Pr =4πr2|ψ|2
Ground state
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This figure shows the three probability distributions for n = 2 and = 1 (the distributions for m = +1 and m = -1 are the same), as well as the radial distribution for all n = 2 states.
Hydrogen Atom Wave Functions
l
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Hydrogen “Orbitals” (electron clouds)
( )2,trrΨRepresents the probability to find a particleAt a location r at a time t
The probability densityThe probability distribution
The Nobel Prize in Physics 1954"for his fundamental research in
quantum mechanics, especially for his statistical interpretation of the wavefunction"
Max Born
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Atomic Hydrogen Radial part
Analysis of radial equation yields:
∞−= RnZEnlm 2
2
chemR e
30
4
8ε=∞with
Wave functions:
( ) ( ) ( ) hr /,, tniElmnlnlm erRtr −Υ=Ψ φθ
( ) ( )( ) ⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
+−−
= +−−
+−
nZL
nZe
nnn
naZu n
nZn
ρρρ ρ 22!12!12 12
1
1/
0
ll
l
ll
naZr /2/ =ρ 220 /4 ea μπε h=
For numerical use:
( )rruR nl=
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Quantum analog of electromagnetic radiation
Classical electric dipole radiation Transition dipole moment
Classical oscillator Quantum jump
2reIrad&&r∝ 2
2*1 τψψ dreIrad ∫∝ r
The atom does not radiate when it is in a stationary state !The atom has no dipole moment
01*1 == ∫ τψψμ driir
20
2
6 hε
μ fiifB =Intensity of spectral lines linked
to Einstein coefficient for absorption:
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Selection rules
Mathematical background: function of odd parity gives 0 when integrated over space
In one dimension: dxxfdxxx ifif ∫∫∞
∞−
∞
∞−
=ΨΨ=ΨΨ )(* with if xxf ΨΨ= *)(
( ) dxxfdxxfdxxfxdxfdxxfdxxfdxxf ∫∫∫∫∫∫∫∞∞∞
∞
∞
∞−
∞
∞−
+−=+−−=+=000
0
0
0)()()()()()()(
0)(20
≠= ∫∞
dxxf )()( xfxf =−if
0= if )()( xfxf −=−
iΨ and fΨ opposite parity
iΨ and fΨ same parity
Electric dipole radiation connects states of opposite parity !
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Selection rules
depend on angular behavior of the wave functions
rr
rr−
θ
φ
Parity operator
( ) ( )πφθπφθ +−→−−−→
−=
,,,,),,(),,(
rrzyxzyx
rrP rr
All quantum mechanical wave functionshave a definite parity
( ) ( )rr rrΨ±=−Ψ
0≠ΨΨ if rr
If and have opposite parityfΨ iΨ
Rule about the mYl functions
( ) ),(),( φθφθ mm YPY ll
l −=
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“Allowed” transitions between energy levels occur between states whose value of differ by one:
Other, “forbidden,” transitions also occur but with much lower probability.
Hydrogen Atom: Schrödinger Equation and Quantum Numbers
l
“selection rules, related to symmetry”
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Selection rules in Hydrogen atom
Intensity of spectral lines given by
ififfi re Ψ−Ψ=ΨΨ= ∫rrμμ *
1) Quantum number nno restrictions
2) Parity rule for l
odd=Δl
3) Laporte rule for l
1r
lr
lr
+= if
Angular momentum rule:
so 1≤Δl
From 2. and 3. 1±=Δl
Lyman series
Balmer series