Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum...

23
Physics 2D Lecture Slides Dec 3 : The End ! Vivek Sharma UCSD Physics

Transcript of Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum...

Page 1: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Physics 2D Lecture SlidesDec 3 : The End !

Vivek SharmaUCSD Physics

Page 2: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

2 22

2 2

2

2

2

2

2

2

m1 sin ( 1) ( ) 0.....(2)sin sin

...............

d ..(1)

1 2m k

e ( 1)(E+ )- ( ) 0....(3)r

Th

m 0..l

l

d R r l lr R r

d d l ld d

r dr r r

θ θθ θ θ θ

⎡ ⎤

⎡ ⎤∂ +⎛ ⎞ + =⎜ ⎟ ⎢

Θ⎛ ⎞ + + − Θ

Φ+ Φ

⎥∂⎝ ⎠ ⎣ ⎦

=⎜ ⎟ ⎢ ⎥⎠ ⎣ ⎦

=

ese 3 "simple" diff. eqn describe the physics of the Hydrogen atom.All we had to do was figure out (with help of French mathematicians) the solutions

Each p

of the

art, c

diff. equat

learly, has

ions

a different functional form and gave clues to behaviour of the Hygrogen atom in its Ground state and excited states

Page 3: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Solutions of The S. Eq for Hydrogen Atom

n = 1,2,3,4,5,....0,1, 2,3, , 4....( 1)

m

: The hydrogen atom is b

0, 1, 2, 3,..Quantum # appear only in Trapped systems

The Spatial Wave Fu

rought to yoIn S u by the letters

ummar

nct

y

.

l

l nl

∞= −= ± ± ± ±

lm

ion of the Hydrogen m

Ato

( , , ) ( ) . ( ) . ( )

Y (Spherical Harmonics)l

l

n l l mm

nl l

r R r

R

θ φ θ φΨ = Θ Φ

=

NOW we are going to take the Hydrogen atom and “ Kick it up a Notch !”

Page 4: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Interpreting Orbital Quantum Number (l)

2

RADIAL ORBITAL

RADIAL ORBI

2 22

2 2 2

22

TAL2 2

K

1 2m ke ( 1)Radial part of S.Eq

K ;substitute this form for

n: (E+ )- ( ) 0r

For H Atom: E = E

K K

K + U =

1 2m ( 1)-2m

d dR r l lr R rr dr dr r

d dR l lrr dr dr r

ker

⎡ ⎤+⎛ ⎞+ =⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

+⎛ ⎞+⎜

+

+⎝ ⎠

[ ]

ORBITAL

RADIAL

OR

2

2

22

B

2

2

( ) 0

( 1)Examine the equation, if we set get a diff. eq. in r2m

1 2m ( ) 0 which

Further, we also kno

depends only on radi

K

K us

w

r of orb

K tha

i

t

t

R r

l l thenr

d dRr R rr dr dr

⎡ ⎤=⎢ ⎥

⎣ ⎦

+

⎛ ⎞+ =⎜ ⎟⎝ ⎠

=

22

ITAL ORBITorb

2

AL 2

2

ORBITAL 22

L= r p ; |L| =mv r

( 1

1 ; K2 2

Putting it all togather: K magnitude of) | | ( 1)2m

Since integer=0,1,2,3...(

A

n-1) angular mome

ng.2

nt

Mom

orbit

l l L l lr

l

Lmvmr

L

posi

mr

tive

× ⇒

+= = +

=

=

= ⇒

=

um| | ( 1)

| | ( 1) : QUANTIZATION OF Electron's Angular Momentum

L l l discrete values

L l l

= + =

= +

pr

L

Page 5: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Magnetic Quantum Number ml

zUncertainty Principle & Angular

(Right Hand Rule)

In Hydr

Momentum : L

A

ogen atom, L can not have precise me

rbitararily picking Z axis as a refer

asurable val

ence directi

ue

L vector s

on:

L r p

φ

= ×

∆ ∆ ∼

Z

Z

Z

The Z component of L

:| L | | | (always)

since

pins around Z axis (precesses).

|L | ; 1, 2, 3...

( 1)

It can never be that |L | ( 1) (breaks Uncertainty Principle)So

l l

l

l

m m l

m l l

Note

m l l

L

= = ± ± ± ±

< +

= +

<

=

you see, the dance has begun !

Dancing in the Dark !

Page 6: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Radial Probability Densities l

2* 2 2

m

2

( , , ) ( ) . ( ) . ( )

( , , ) |

Y

Probability Density Function in 3D:

P(r, , ) = =| | Y |

: 3D Volume element dV= r .sin . . .

Prob. of finding parti

| | .

cle in a ti

n

l

l

l

nlnl lml

n

m

mll

Note d

r R r

r

r d

R

d

R

θ φ θ φ

θθ φ

φ

φ

θ θ

Ψ = Θ Φ =

ΨΨ Ψ =

l

l

2 2

22

m0

2

2

0

2 2

m

y volume dV isP.dV = | Y | .r .sin . . .The Radial part of Prob. distribution: P(r)dr

P(r)dr= | ( ) |

When

| ( ) |

( ) & ( ) are auto-normalized then

P(r)dr

| | .

| |

=

.

|

l

l

l

lm

l

m

m

n

nl

n

llR

R r d

R

dr

d

d

r

d

dπ π

θ

θ φ

φ φ

θ φ

θ

θ

Θ

Θ

Φ

Φ

∫ ∫

2 2

2 2nl

2 2

0

0

in other words

Normalization Condition: 1 = r |R | dr

Expectation Values

P(r)=r |

<f(

| |. . ;

r)>= f(r).P(r)dr

nll r r Rd

Page 7: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Ground State: Radial Probability Density

0

0

0

0

0

2 2

2

2

230

0

30

2

0

2

2

( ) | ( ) | .4

4( )

Probability of finding Electron for r>a

To solve, employ change of variable

2rDefine z= ;

4

1

limits of integrationa

2

ra

r a

r

ra

a

a

P r e dr

P r dr r r dr

P r dr r ea

c

a

P

han

z

ge

ψ π

∞ −

>

>

=

=

=

⎡ ⎤⎢ ⎥⎣ ⎦

=

2 22

(such integrals called Error. Fn)

1 66.7%! =- [ 2 2] | 5 0.66 !72

z

z

z z e

e z

e

d

− ∞

∞−

+ + = = ⇒

Page 8: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Most Probable & Average Distance of Electron from Nucleus

0

0

22

30

222

30 0

4In the ground state ( 1, 0, 0) ( )

Most probable distance r from Nucleus What value of r is P(r) max?

dP 4 2=0 . 0 2dr

Most Probable Distance:ra

l

rad

n l m P r dr r ea

rr e ra dr

ea

= = = =

⎡ ⎤ ⎡ ⎤−⇒ ⇒ = ⇒ +⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦0

0

2

2

00

022

30

0

0

2 2 0 0 ... which solution is correct?

(see past quiz) : Can the electron BE at the center of Nucleus (r=0)?

4( 0) 0 0! (Bohr guessMost Probable distance ed rig

ra

a

r r r or r aa

P r e r aa

=

⇒ + = ⇒ = =

= == = ⇒

02

230r=0 0

3 n0

0

0

0

ht)

4<r>= rP(r)dr=

What about the AVERAGE locati

. ...

on <r> of the electron in Ground state?

2rcha

....... Use general for

nge of variable

m

z=a

z ! (4

z z

z

rar r e

ar z e dz e dz n n

d

n

ra

∞ ∞ −

∞ ∞− −

=

⇒< >= = =

∫ ∫

∫ ∫

0 00

1)( 2)...(1)

3 3! ! Average & most likely distance is not same. Why?4 2

Asnwer is in the form of the radial Prob. Density: Not symmetric

n

a ar a

− −

⇒ < >= = ≠

Page 9: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Radial Probability Distribution P(r)= r2R(r)

Because P(r)=r2R(r) No matter what R(r) is for some n

The prob. Of finding electroninside nucleus =0

Page 10: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Normalized Spherical Harmonics & Structure in H Atom

Page 11: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Excited States (n>1) of Hydrogen Atom : Birth of Chemistry !

200

n 211 210 21-1

3/ 21

211 21 10 0

Features in & : Consider Spherically Symmetric (last slide)Excited States (3 & each same E ) : , , are all states

1 Z=R Y = a

2, 0 2p

8Z r

a

n lθ φ

ψψ ψ ψ

ψπ

⇒ =

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

=

⎝ ⎝

=

⎠0

211

0 02

2 * 2211 211

10 21 1 1

z

1 3(r) Y ( , );Y ( , ) cos ; max at =0, min at =2 2

We call this 2p state because of it

sin

| | | | sin M

s extent

ax at = , min at =0; Symm

in

. .

in 2

z

ra iZ

e

R

e φ

πψ θ

θ

πψ ψ ψ θ θ θ

φ θ

φ

φ θ θ θπ

= ∝

=

2pz

22

'1 2

'

1 2Principle of Linear Superposition: If & are sol. of TISE

then a "designer" wavefuntion is

Remember,

also a s

for a TISE: -2

ol. of T

m

To check this, substitute in place o

ISE

f

a b

U E

ψ ψ

ψ ψ ψ

ψ

ψ ψ ψ

∇ +

= +

=

22 ' ' '

& convince yourself that

-2m

U Eψ

ψ

ψ ψ∇ + =

Page 12: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Designer Wave Functions: Solutions of S. Eq !

[ ]

[ ]

x

y

2p 211 21 1

2p 211 21 1

Linear Superposition Principle means allows me to "cook up" wavefunctions1 ......has electron "cloud" oriented along x axis2

1 ......has electron "cloud" oriented along2

ψ ψ ψ

ψ ψ ψ

= +

= −

200 210 211 21 1 2 , 2 , 2 , 2

Similarly for n=3 states ...and so on ...can get very complicated structure in & .......whic

y axis

So from 4 solutio

h I can then mix & match

ns

to make electron

, ,

,

s "

x y zs p p p

θ φ

ψ ψ ψ ψ − →

most likely"to be where I want them to be !

Page 13: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Designer Wave Functions: Solutions of S. Eq !

Page 14: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Energy States, Degeneracy & Transitions

Page 15: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

The “Magnetism”of an Orbiting ElectronPrecessing electron Current in loop Magnetic Dipole moment µ

2A rea o f cu rren t lo

E lectro n in m o tio n aro u n d n u cleu s circu la ting charg e cu ren t

; 2 2

-eM ag netic M o m en t | |= i

o p

A = ; 2 m

L ike the L , m agn eti

A = r

-e -e 2 m 2 m

c

ie e epi rT m r

v

r r pp L

π

µ

π π

µ

⇒ ⇒− − −

= = =

⎛ ⎞⎜ ⎟

⎛ ⎞ ⎛ ⎞= × =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎝⎠ ⎠

z-e -ez com po nen t, !

2

m om en t a lso p rece

m

sses abo u t "z" ax i

m

s

2z l B lL m m quan tized

µ

µ µ⎛ ⎞ ⎛ ⎞= = = − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Page 16: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Quantized Magnetic Moment

z

e

-e -e2m 2m

Bohr Magnetron

e =2m

z l

B l

B

L m

m

µ

µµ

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −=

⎛ ⎞⎜ ⎟⎝ ⎠

Why all this ? Need to find a way to break the Energy Degeneracy & get electron in each ( , , ) state to , so we can "talk" to it and make it do our bidding

identify it: "Puppy tr

selfaining"

ln l m

Page 17: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

“Lifting” Degeneracy : Magnetic Moment in External B Field

Apply an External B field on a Hydrogen atom (viewed as a dipole)

Consider (could be any other direction too) The dipole moment of the Hydrogen atom (due to electron orbit)

experi

B ||

e

Z axis

Torque which does work to align ||but this can not be (same Uncertainty principle argument)

So, Instead, precesses (dances) around ... like a spinning

nces

topT

a

he Azimuthal angle

B B

B

τ µ µ

µ

= ×

L

|projection along x-y plane : |dL| = Lsin .d|dL| ; Change in Ang Mom.

Ls

changes with time : calculate frequencyLook at Geometry:

| | | | sin2

d 1 |dL 1= = = sindt Lsin dt Lsin 2

inqdL dt LB dtm

q LBm

d

qB

θφ

τ θ

ωθ

φ

φ

φθ

θ

θ

=⇒ = =

⇒ =

L depends on B, the applied externa

Larmor Freq2

l magnetic f l ie dem

ω

Page 18: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

“Lifting” Degeneracy : Magnetic Moment in External B Field

WORK done to reorient against field: dW= d =- Bsin d

( Bcos ) : This work is stored as orientational Pot. Energy U

Define Magnetic Potential Ene

dW= -

rgy U=- .

dU

B

d d

B

W

µ τ θ µ θ θ

µ θ

µ

=

=

e

cos .eChange in Potential Energy U =

2m L

z

ll

B

m B m

Bµ θ µ

ω

− = −

=

Zeeman Effect in Hydrogen Atom

0

In presence of External B Field, Total energy of H atom changes to

E=E

So the Ext. B field can break the E degeneracy "organically" inherent in the H atom. The E

L lmω+

nergy now depends not just on but also ln m

Page 19: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Zeeman Effect Due to Presence of External B fieldBye Bye Energy Degeneracy

Page 20: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

Electron has “Spin”: An additional degree of freedom

Page 21: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

The Consequence of Spinning Electron

Page 22: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum
Page 23: Physics 2D Lecture Slides Dec 3 : The Endmodphys.ucsd.edu/4Es04/Slides/F03/Dec3.pdfMagnetic Quantum Number m l Uncertainty Principle & Angular z (Right Hand Rule) In Hydr Momentum

The Periodic Table based in 4 Quantum numbers