2.1 The particle in a box 2.1.1 The Partical in a One ... · • nof , ¸2È ^1ç c: ` 5¤ 0)æ,º...

Nankai University http://struchem.nankai.edu.cn

Quantum Mechanics of Some Simple Systems


2.1 The particle in a box

2.1.1 The Partical in a One-Dimensional Box

I III V= (x 0, x l )

0 l x

V(x) III III

0 l x

V(x) III III2 2

2

( ) ( ) ( )2

d x x E xm dx

2 2

2

( ) ( ) ( ) ( )2

d x E x xm dx

2

2 )(1)(dx

xdx

I(x) = 0 III(x) = 0


II ( ) cos sinx A kx B kx 2k mE

0 0I IIlim limx x A = 0

III IIlim limx l x l

Bsinkl = 0B 0

kl = n , n = 0,1,2…n 0

2 2

2 1,2,38n hE nml

II V=0 ( 0< x < l )2 2

2

( ) ( )2

d x E xm dx

0 l x

V(x) III III

II ( ) sin n xx Bl

0 2 222 2 2 2I II III0 0

sin 12

l l

l

ldx dx dx dx B n x l dx B

2B l 2 iB le=0

2B l II2 sin n xl l


8ml 2h2

E1=

E2= 4E1

E3= 9E1

E4= 16E1

n = 1

n = 2

n = 3

n = 4 --

-

-

++

++

+

+

l0l0

| (x)|2 (x)

x

x

II2 sin n xl l

2 2

2 1,2,38n hE nml

• n• n• ,

(zero-point energy)• E=En+1 En=(2n+1)h2/8ml2 l E 0


l/2

n = 1

n = 2

n = 3

n = 4--

-

-

++

++

+

+

l0

(x)

(x)R

ˆ ( ) ( )R x x l x=l/2

2 2 2 2 2 2

2 2 2ˆ ˆ ˆ ˆ

2 2 ( ) 2d d dRT R T

m dx m d x l m dxˆ ˆ ˆRV V

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( )RH R T V RT RV T V H

ˆ ( ) ( )H x E x ˆ ˆ ˆ ˆ ˆ( )[ ( )] ( )[ ( )]RH R x RE R x

ˆ ˆ ˆ[ ( )] [ ( )]H R x E R x

ˆ ( ) ( )R x c x2 2

0

ˆ ( ) 1l

R x dx c c = 1


R (symmetric), (even)R (antisymmetric), (odd)

1 3 2 4

0

0( ) ( )

1

l

m n

m nx x dx

m n

l/2

n = 1

n = 2

n = 3

n = 4--

-

-

++

++

+

+

l0

(x)


2.1.2 Partical in a two-dimensional well

O

ax

b y

8

88

8

V(x, y)

),(),(2 2

2

2

22

yxEyxyxm

)()(),( yYxXyx

22

2

2

2 2)()(

1)()(

1 mEy

yYyYx

xXxX

22

2

22

2

2)()(

1

2)()(

1

y

x

mEdy

yYdyY

mEdx

xXdxX E = Ex + Ey


xa

na

xX xsin2)( 2

22

8mahnE x

x

2

22

8mbhn

E yy

nx=1, 2, 3…

ny=1, 2, 3…

byn

axn

abyx yx sinsin2),(

2

2

2

22

8 bn

an

mhEEE yx

yx

nx=1, 2, 3… ny=1, 2, 3…

+

O

a

bb

a

O

+-

O

a

b

+

-+

+-

-

O

a

b

1,1 1,2 2,1 2,2

2( ) sin ynY y y

b b


czn

byn

axn

abczyx zyx sinsinsin22),,(

2

22

2

22

2

22

,, 888 mchn

mbhn

mahn

E zyxnnn zyx

nx=1, 2, 3… ny=1, 2, 3… nz=1, 2, 3…

a = b = c = l

0

5

10

2,2,2

1,1,31,3,13,1,1

1,2,22,1,22,2,1

1,1,21,2,12,1,1

E/(h

2 /8m

l2 )

1,1,1

degeneracy

2.1.3 Partical in a three-dimensional box


2,2,2

3,1,1 1,3,1 1,1,3

2,2,1 2,1,2 1,2,2

2,1,1 1,2,1 1,1,2

1,1,1


2.1.4 The free partical in one dimension

V(x)=0 Schrödinger2 2

2

( ) ( )2

d x E xm dx

2

2 2

( ) 2 ( ) 0d x mE xdx r2+k2=0

1 2( ) ikx ikxx c e c e

2mEk

r = ik( ) cos sinx A kx B kx

E<0 2 2 0ik i i m E m E

x + e-ikx +x eikx + E 0

(x)


1 1( ) ikxx c e

11 1 12 ikx

x xdp i mEc e pdx

1 2 H xp

1 1 1xiE t

x xie c exp p x E t

1 +x +|px| 2 x|px|

1 x x+dx

1 1 1 1 1 1* * *dW dx dx c c dx =c1

*c1=

xpx x

2 2( ) ikxx c e

2 2x xp p

2 2 x xic exp p x E t

1 2( ) ikx ikxx c e c exp


2.2 Penetration into and through barriers

2.2.1 An infinitely thick potential wall

0 x

V

I II

I (x<0)2 2

2ˆ

2dH

m dx

II (x 0)2 2

2ˆ

2dH V

m dx

Iikx ikxAe Be 2k mE

IIik x ik xA e B e 2 ( )k m E V

E<V x>0


II IIik x ik xA e B e 2 ( )k m E V

E<V k' k'=i

IIx xA e B e 2 ( )m V E

x e x

II = A' e x

• II

• 1/ penetration depth


2.2.2 A barrier of finite width

0 l x

V

I II III

Zone I (x<0): V(x)=0Zone II (0 x l): V(x)=VZone III (x>l): V(x)=0

Iikx ikxAe Be 2k mE

IIik x ik xA e B e 2 ( )k m E V

IIIikx ikxA e B e 2k mE

A. E > V

• I Aeikx Be ikx

• II A'eik'x B' e ik'x

• III A"eikx B"=0

I(0) = II(0) A+B=A'+B'

I II

0 0x x

d ddx dx

kA kB=k'A' k'B'


II(l) = III(l)

II III

x l x l

d ddx dx

ik l ik l iklA e B e A eik l ik l iklk A e k B e kA e

2 2

4( ) ( )

ikl

ik l ik l

kk eA Ak k e k k e

2 2

2 2

2 ( )sin( ) ( )ik l ik l

i k k k lB Ak k e k k e

2D

kJ Am

2R

kJ Bm

2kJ Am

2 2

2 2 2 2 2

4( )sin ( ) 4

DJ k kDJ k k k l k k

1RJR DJ

l

V

x0


B. E < V

k' k'=i

2 2 2

2( ) sh( ) 2 ch( )

iklik eA Ak l ik l

2 2

2 2 2 2 2

4( )sh ( ) 4

kDk l k

1l 2 2sh ( ) ( ) 2 4l l ll e e e

2 2 ( )2l m V ElD e e

(l ) (V )m


2.3 The harmonic oscillator

2 22

2

1ˆ2 2

dH kxm dx

2 2

2 2 2

2 0d mE mkxdx

=2mE/ 2 mk2

2 22 0d x

dxz x

22

2 0d zdz

z z2>> / d2 /dz2 = z2

2 2ze2 2 2

22 2 2 2 2

2 ( 1)z z zd e e z z edz

2 2ze2

22 0d z

dz2 2( ) ( )zz e U z

z ,


2

2 2 1 0dU dUz Udz dz

0

kk

kU a z

1

0

kk

k

dU a kzdz2

220

( 1)( 2) kk

k

d U a k k zdz

20 0 0

( 1)( 2) 2 1 0k k kk k k

k k ka k k z a kz a z

z, , zk 0: ak+2(k+1)(k+2) + ( / 1 2k) ak= 0

22 1( 1)( 2)k k

ka ak k

k = 0, 1, 2, …,

/ k UU / =2n+1 (n = 0,1,2,…)

an+2=an+4=…=0 U


2 2 1m E nk

1(2 1) ( )2 2

kE n n hvm

(n = 0,1,2,…)

2

2 2 2 0dU dUz nUdz dz

Hermite

0( ) ( )

nk

n kk

U z H z a z 22( )

( 1)( 2)k kn ka a

k k

2 2( ) ( )zn n nx N e H z z x

2

21

212

1

)(!2

1)(x

nnn exHn

x

H0(z)=1 H1(z)=2zH2(z)=4z2 2 H3(z)=8z3 12zH4(z)=16z4 48z2+12 H5(z)=32z5 160z3+120z

zHn = ½Hn+1 + nHn 1


n=3

n=2

n=1

x

V(x) (x)

n=0

E3=7hv/2

E2=5hv/2

E1=3hv/2

E0=hv/2

| (x)|2 1( )2

E n hv

2

21

212

1

)(!2

1)(x

nnn exHn

x

T = V

•

• x=0

•

•


2.4 Partical on a ring

2 2 2

2 2ˆ

2H

m x ym xy

r 0

x=rcosy=rsin

2 2 2

2 2 2

1 1ˆ2

Hm r r r r

r2 2 2 2

2 2 2ˆ

2 2d dH

mr d I d

2

2 2

2d IEd

l lim imAe Be 2lm IE

( )= ( +2 )

2 cos( 2 ) sin( 2 ) 1liml le m i m

2 2

2l

lm

mEI

ml=0, 1, 2,…


ˆzM i x y i

y x

B=0 ˆl lz m l mM ml

l

imm Ae

2 2*

02 1d A

1 2 1 2(2 ) (2 ) cos sinl

l

imm l le m i m

cos

sin

( ) 1 cos

( ) 1 sinl

l

m l

m l

m

m

ml = 0

ml = 1

ml = 2


2.5 Partical on a sphere

x=r sin cosy=r sin sinz=r cos

2 2 22 2

1 1rr r r r

22

2 2

1 1 sinsin sin

r2 2

2 22

ˆ2 2

Hmr I

22

2E

I2

2

2IE k

( , ) ( ) ( ) 1 2( ) (2 ) l

l

imm e ml=0, 1, 2,…

2

2

1 sin 0sin sin

lm k z = rcos P(z)= ( )

sind dP dz dPd dz d dz


22

2(1 ) 01

lmd dPz k Pdz dz z

22( ) (1 ) ( )lmP z z G z

2(1 ) 2( 1) ( 1) 0l l lz G m zG k m m Gn

nn

G a z

2( 1)( 2) ( 1) 2 ( 1) ( 1) 0n l l l nn n a k m m n m n n a

2

( 1) 2 ( 1) ( 1)( 1)( 2)

( )( 1)( 1)( 2)

l l ln n

l ln

k m m n m n na a

n nk n m n m

an n

( )( 1)l lk n m n m G G z= 1


l=n+|ml| l=|ml|, |ml|+1, …

k = l(l+1)

2 ( 1)l l

1 2

2( )!2 1( ) ( 1) (cos )

2 ( )!l

m mml

ll

l ml Pl m

Associated Legendre functions

,( , ) ( ) ( )l l llm l m mY

Spherical harmonics

l = 0,1,2,…ml= l, l 1, …, l

l ml Ylml( , )

0 0 1/2/21 0 ½(3/ )1/2cos

1 (3/2 )1/2sin e i

2 0 ¼(5/ )1/2(3cos2 1)1 ½(15/2 )1/2cos sin e i

2 ¼(15/2 )1/2sin2 e 2i

3 0 ¼(7/ )1/2(2 5sin2 )cos1 (21/2 )1/2(5cos2 1) sin e i

2 ¼(105/2 )1/2 cos sin2 e 2i

3 (35/ )1/2sin3 e 3i

2

( 1)2llmE l l

I 2l+1

x

y

+

+Real component

Imaginary component

l=1, ml= 1

Condon–Shortley phase


2

1

0

-1

-2

z

ml

6

6

6

6

6

I: , :212

E I

l = IE = l2/2I

= ( 1)l llangular momentum quantum number

z

ˆ2

l

l l

im

z lm lm leM Y m

i


1 2

2( )!2 1 1( , ) ( 1) (cos ) exp( )

2 ( )! 2

(cos )exp( )

l

l

m mml

lm ll

ml

l mlY P iml m

NP im

2(2 1)( )!

( 1)4 ( )!

m ml

l

l l mN

l m

( )

3 sin83 sin

8

i

i

p e

p e

1 3( ) sin cos423( ) sin sin

42

x

y

p p p

ip p p

Condon–Shortley


2.6 The rigid rotor

Rm1

m22 2 2

2 2

1 2

1 1ˆ2 2 2 cmH

m m m

m=m1+m2

1 2

1 1 1m m

2 22 2

2 2cm totalEm

cm

22

2 cm cm cm cmEm

22

2E

22

22E

R

2I R2

( 1)2JJME J J

I


2.7 Hydrogenic atoms2 2 2 2 2 2

2 2 2 2

0 0

ˆ2 2 4 2 2 4e N cm

e N

Ze ZeHm m r m r

2 22

02 4Ze E

r2

2 22 2 2 2

0

1 1 22

Ze Err r r r r

( , , ) ( ) ( , )r R r Y2 2

2 2 2 20

1 ( ) ( 1) 22

d rR Ze l l ER Rr dr r r

u=rR2 2 2

2 2 2 20

2 ( 1) 24 2

d u Ze l l Eu udr r r


2 2 2

2 2 2 20

2 ( 1) 24 2

d u Ze l l Eu udr r r

2

20

24Zea b=l(l+1)2

2

2 E

22

a bu u ur r

E<0

r 2u u ru e

ru e( ) ru L r e

22 0a bL L Lr r

( ) nn

nL r c r

2 1( 1) (2 ) 0n nn

nc n n b r n a r

1(2 )( 1)n n

n ac cn n b

2n =a4 2

2 2 208e ZEh n


associated Laguerre functions2 12

1( ) ( )l lnl nR r Ce L

2 1 12 1 1

1 2 1 1( ) ( )l n

l nn l n

d dL e ed d 0

2Z rna

20

0 2

4

e

am e

Bohr radius

3/2 /21,0 0( ) 2R r Z a e

3/2 2 /23,0 0

1( ) 6 69 3

R r Z a e

3/2 /22,0 0

1( ) 22 2

R r Z a e 3/2 2 /23,1 0

1( ) 49 6

R r Z a e

3/2 /22,1 0

1( )2 6

R r Z a e 3/2 2 /23,2 0

1( )9 30

R r Z a e


0 5 10 150.0

0.2

0.4

0.6

0.8

r/a0

2s

0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

r/a0

1s

0 5 10 15 20 25

-0.1

0.0

0.1

0.2

0.3

0.4

r/a0

3s

0 5 10 15 20 25 30 35

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

r/a0

3p

0 5 10 150.00

0.05

0.10

0.15

r/a0

2p

0 5 10 15 20 25 30 350.00

0.01

0.02

0.03

0.04

0.05

r/a0

3d


0 1 2 3 40.0

0.2

0.4

0.6

r2R210

0 5 100.0

0.1

0.2

r2R220

0 5 100.0

0.1

0.2

r2R221

0 10 200.00

0.05

0.10

r2R230

0 10 200.00

0.05

0.10

r/a0 r/a0

r2R231

0 10 200.00

0.05

0.10

r/a0

r2R232

22 2 2

0 02 2

( ) sinlnl lmD r dr R Y r drd d

R r


2py2px2pz

1s 2s


3py3px3pz3s

4py4px4pz4s


3dz2 3dx2-y2

3dxy 3dxz 3dyz


4dz2 4dx2-y2

4dxy 4dxz 4dyz


4fz(x2-y2) 4fzxy 4fx(x2-3y2) 4fy(3x2-y2)

4fz3 4fxz2 4fyz2

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2.1 The particle in a box 2.1.1 The Partical in a One ... · • nof , ¸2È ^1ç c: ` 5¤ 0)æ,º...

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