6. quantum mechanics 2, 5 and 7. hydrogen atoms, 1schroedinger •move much?!1914 vienna...
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6. Quantum Mechanics 2, 5 and
7. Hydrogen Atoms, 1
lecture 25, October 25, 2017
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housekeeping
Exam 2
Next Friday, October 27
Thornton and Rex, Chapters 3,4,5, 6.1-6.3
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today
Simple Harmonic Oscillator
Hydrogen atom
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Schroedinger
•move much? ! 1914 Vienna Habilitation, WWI, 1920 Jena, 1920 Stuttgart, 1921 Breslau,
1921 Zurich, (TB treatment 1923), 1927 Berlin, 1934 Oxford, 1933 Nobel, 1934 Princeton? nope, Edinburgh? nope, 1936 Graz…nope, 1933 Oxford, Ghent, Dublin, 1955 Vienna, 1961 died.•1925 ! learned of deBroglie’s work •1926: ! January: Quantization as an Eigenvalue Problem: Hydrogen! February: harmonic oscillator, rigid rotator, diatomic molecules! May: equivalence with Heisenberg and Stark Effect
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Quantum Oscillator
aka simple Harmonic oscillator,
SHO
In general " F= - hx classically .
quantum mechanics ?
not " forces"
but potentials , right ? obey : E= TTV →
F =-
d¥
V
§§'¥#§§interim : vw=%n×
.
KK )
• ↳ V ( x )recipesTK ) → se.
T
£Classical turning points
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them dglxkd+ ±hx24Cx ) = EXG )
KIKI = -
off( E -
tzhx74K)=( ahthx
'
-
Inge)4x )
series solution :
variable substitution. . y= 4✓mgI × = xx ; y2=x2×2
e =
2¥ JMI
xez : du =L dx, dg2=x2dE
d4- = ( y
'. e) 4
dy2
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dey
,
= ( y'
- E) 4 Gsnmtoticaun :
y'
> , E
d24
In= g- 4
± 4212+ 4 = e
⇒ want only - sign to keep
4 finite as yen
So solution looks like
4cg ) = C I"%
Hcy )(waggaman g. ↳ way
substitute
get auxiliary equationfor H ( y ) →
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DIED-
zydattg+ ( E - DH =o *
standard approach : assume solution
r
�1� HC g) = ao + a. y + azy2+ . . . = {oany
" )substitute
d¥y
=
ait2asyt3a3y2t.dTyIz-2azt6a3ytlzaay2t.i-EoCntDCntDannyn@24dtqt-2ya.t
4azy2 + . . . = Zznany
"
@
�1�,
@,
@ - *
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Foy "[ ( nti )(u+z)an+z - znant ( E . 1) an ] =o
only true when [ .-
- ]=o ⇒ an+z= 2M¥ an
anti ) ( ntz )
a RECURSION FORMULA
while HCY ) in terms
ofao & a
, . - and generate the rest
a
normalization : f 4*44=1 only if series terminates
- °
at some h .
Conditions :
l,
For some Shui fire n, au+z= 0 which happens
zntl = E =
2zEJmI
2. If that his odd
,ao=o ;
if even 9=0
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condition 1.
zntl = E =
2¥ mzf
auantize¥a=( at E) tha
In an oscillator...
f.=§n zt
,=
VERY !
En=(htlz)hf
VERYa
BIG DEAL
condition 2.
⇒ particular setof polynomials , Hucy )
Hermit Polynomials
even functions if n even
ll ✓ ( UL
odd odd
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very familiar throughout pmsics
such polynomial sets are produced from"
generating functions"
Hermila scy ,s ) = I
52+24= ¥
oHunt.ms
"
or
Hly ) = ( - , )"
e-42 ICI )
dy"
Restore constants :
Ynlx ) = ⇐znn ,)± of "%Hn*termite Polynomials
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Here are some : &= FudaHo ( x x ) = 1
Hi( Lx ) = Zxx
Hzcxx ) = 4×2×2-2Eu=(httz)hf
Has ( xx ) = 8×3×3 - lzxxEn
= @t Yz)Tµ
Hqlxx ) = 16×4×4 - 482×2+12
So fn first 2 :
Yz -
&2X%
.
Hold= ( Ff ) e Eo=
tzhw←
Zero point
energy . . again
- dxYz
Yi ( x ) =⇐#)k2××e E. =
3zKw
MATHEMNNCA →
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r=FIw.
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VERY USEFUL- -
'
nearly
Many potentials avenpavchohl
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RELATIVISTIC QUANTUM FIELD THEORY
It's an fields. . .
"
particles"
are harmonic excitations of the parent
quantum field
cartoon of X field
- T
€¥¥tfmIMitraitem.
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HYDROGEN
Ato±
g.F
✓ H= -
qfz.
e÷ →F → s
. E
.
i.( '
44-
'
.
(,
- -4.1X
"
Need spherical coordinates "
0 = cost ( Tf )
"
polar angle
"
4=4= tan'
( ¥ )"
azimuthal angle"
z= r cos 0
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electron
=
( MI ) me4 (F) = 4 ( ×, y . z ) n
→ reduced mass
me + mp
- It [ ¥ .
+ Eu + 3÷g ] 44%3 ) + VK.u.zhkxu.gl= EXG , a. g)
:h±µtf24C
x. yiz ) + Vcx, %z)4( x. a. 3)
= E 44. g. 3)
Laplacian Operator ( Cartesian )
Need spherical coordinates ! → Lapcacian changes. . .
transforms
172 = trz Er Cryer ) + resting 3fGm0§o ) +
rtzsneo Fp
5. E .
zEfp24(v. a ¢ ) +
Yr) 4 ( no , 9)
= E4( v. a. a)
¥<
HE or
Need to solve this. -
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In spherical coordinates,
solution is separable :
4C F) = 4 ( r,
a, d) = RCDT ( a) Icq )
separation of variables technique to solving ( many! )
differential equations in physics
IE Hegarnered)+ nstno IotsmoZones ) +
rtzswzoFaiths ]( qq.mg, = . µ ,
a
substitute and calculate
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4C F) = 4C no ,
d) = RCDH a) Icq )
If[ tn Eraser
#
5)+nstnossocsmostfkisltrtzswzoFailed ¥2.
,
# = Excel
.
substitute. .
the§r
's act only on Rcr )
2is
"
TCG )
:is
"
I (9)2g
so,
substitute and divide by RCMTC E) Ice ) ,multiply by
-
2ftr2sin4
horn at
finto see :
zh±µ[f. Ir(
m§r4Ct)
)]→zh±⇐.§r(mzglrcrlttotpca ) ) )
µ
= -52
how divide :In#" 0 # 4) Er
'
#Rcrj )=T±⇐, IEEE'a£rt£rrcn]÷E⇐
trarrttertyµ
now multiply :
=
5y÷0§rm[ IF ]
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fieryErhardt
sett
£ocsioaEoDtEud¥a9
'
÷÷fm4( fear
+ e)To
-4is only in the 3rd term so this can be written
- [ ] . [ ]=
eta, "¥a±
'
=-
no 9 dependence
↳ r.tt dependence
= A Constant . ( a separation
E- m2 ( NOT a mass
! )
- stant"
)
¥( g) d¥fl= . mz satin:
e( a)=eim9
.
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-
gspeaojErhardt
sniff,
Ioc swept
Ea , diedin
'
onfadefth.CI#+eD=in r - land → divide -
out
team@+
2¥'
( Ear +
e)=
tea ntasuioatotsmodattoo' )
--
only r dependence only 0 dependence
so,
same song ,
different verse " set = a common constant
=
In
Result : 2
equations.