6. Quantum Mechanics 2, 5 and

7. Hydrogen Atoms, 1

lecture 25, October 25, 2017

housekeeping

Exam 2

Next Friday, October 27

Thornton and Rex, Chapters 3,4,5, 6.1-6.3

today

Simple Harmonic Oscillator

Hydrogen atom

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

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

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

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 ) →

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�,

@,

@ - *

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

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

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

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 →

r=FIw.

VERY USEFUL- -

'

nearly

Many potentials avenpavchohl

RELATIVISTIC QUANTUM FIELD THEORY

It's an fields. . .

"

particles"

are harmonic excitations of the parent

quantum field

cartoon of X field

- T

€¥¥tfmIMitraitem.

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

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. -

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

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 ]

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

.

-

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.