Chem 110A Equations for Exams

Operators

 

ˆ                                         2 2

ˆ        2  2

 

1D:

 

3D:

p= –ix                  KE =- 2m  x2

pˆ = – iÑ

H =-

2m

 x2 + V (x)

 

 

 

3D volume element:

dV = dxdydz = r 2 sinJdrdJdj

 

Ñ2 = 1

r2   +  1       

sinJ   +      1      

2

 

r2 r

r   r2 sinJ ¶J

¶J    r2 sin2 J ¶j 2

 

Schrödinger equation:

Hˆ Y= EY , time-dependent: Hˆ Y(t) = i ¶ Y(t)

t

 

Uncertainty products: Da Db ³

éAˆ, Bˆù

where both operators are Hermitian.

 

ë       û

 

 

=     Y(t) éHˆ, Aˆù Y(t)   when operator Aˆ

itself does not depend on time

 

dt                            ë       û

 

 

Constants and Units

h

E = 27.211 eV = 4.3597 ´1018 J                                                      mass of electron

e

m = 9.109 ´10-31kg

 

h = 6.626 ´10-34 J s                                                                  = h / 2p = 1.055´10-34 J s

0

= 5.292 ´1011m = 0.5292 Å                                                         charge of electron e = -1.602 ´1019 C

 

atomic units:  = me

= a0

= e /

= 1,        atomic unit of time = 24.189 ´10-18s

 

            1             4pe2

e

fine structure constant: a =             =                                                         Bohr radius = a0  = 0                                                      

 

me c a0

137.036

m  e2

 

 

 

y (               æ npx ö

n2p 22

n2h2

 

Particle in a 1D box:

n x) =

2

sinè L

d2     k

ø                        En =

2

2mL2 = 8mL2

m m

 

Harmonic Oscillator:  H =-         +                                                             x

 

m =  1  2

 

 

 

æ a ö

1/ 4

y 0 ( x) = ç p ÷

ea x2 /2

2m dx 2    2

m1 + m2

 

è    ø      E = æ v + 1 ö w

 

3 1/ 4

v        ç      2 ÷

 

y ( x) = æ 4a ö

xea x2 /2                                                                              è                                                               ø

 

1                 ç p ÷

 

è        ø

1/ 4

with                w =

 

y ( x) = æ a ö

(2a x2  -1)ea x2 / 2

 

4p

2                  ç      ÷

è      ø

3 ( )    9p

æ a 3 ö1/ 4

y    x  = ç      ÷

è      ø

(2a x3

  • 3x)e

 

 

a x2 / 2

æ km ö1/ 2

a =

ç   2  ÷

è      ø

 

Hydrogen and other one-electron atoms

 

Z 2 æ

e2      ö        Z 2              Z 2

 

En =-   2 ç

÷ =-   2  Eh     in atomic units:  En  =-      2

 

 

2è a0 4pe0 ø      2n                        2n

 

æ

Angular momentum operators, Spherical Harmonics and Spin angular momentum

ˆ2             2         1             1     2 ö

 

L = -

ç sinJ ¶J sinJ ¶J + sin2 J ¶j 2 ÷

 

è ø

Lˆx  = ypˆz  zpˆ y Lˆy  = zpˆ x  xpˆz Lˆz  = xpˆ y  ypˆ x

 

 

 

ˆ                1 ˆ2

 

Sˆ2   a

= 3 2 a

4

Sˆz   a

= 1  a

2

 

H rigid rotor = 2I L

Sˆ2   b

= 3 2 b

4

Sˆz    b

=- 1  b

2

 

 

Hydrogenic Radial Functions, Rn,l (r)                                                              (Z = atomic number)

 

æ Z ö3/ 2

1 æ Z

ö3/ 2          –s

 

R1,0

(r ) = 2 ç     ÷

a

es

R2,1

(r ) =      ç       ÷

2a

o e 2

 

è   0 ø

æ  Z

ö3/ 2 æ

o ö –s

è

4 2 æ

0 ø

Z  ö3/ 2       æ

o ö  –s                            Zr

 

R2,0 (r ) = 2 ç       ÷    ç1-

÷ e 2

R3,1 (r ) =

ç       ÷    s ç1-

÷ e 3              s =

 

è 2 a0 ø    è      2 ø

9 è 3a0 ø

è      6 ø                     a0

 

æ Z ö3/ 2 æ

2s     2s 2 ö –s

2 2 æ Z

ö3/ 2            –s

 

R    (r ) = 2 ç       ÷    ç1-      +

÷ e 3

R    (r ) =          ç       ÷

o 2e 3

 

3,0

è 3a0 ø    è

3       27 ø

3,2

27 5 è 3a0 ø

 

 

 

sin2æ a ö = 1 cosa

Trigonometric identities

cos2æ a ö = 1+ cosa

 

ç   ÷                                          ç   ÷

è 2 ø          2                              è 2 ø                2

sin x sin y 1 cos(x – y) – 1 cos(x y)

2 2

cos x cos y = 1 cos(x y) + 1 cos(x + y)

2 2

sin x cos y = 1 sin(x y) + 1 sin(x + y)

 

 

e± ix

 

= cos x ± i sin x                sin x =

2

 

eix – eix

 

2i

2

 

cos x =

eix eix

 

2

 

Integrals

 

 

 

ò sin2(ax)dx = x  1 sin(2ax)

Indefinite Integrals

ò sin(ax)cos(ax) = 1 sin2(ax)

 

2     4a

x      1

2a

m               r        mxm – r

 

ò cos2(ax)dx =

+

2     4 a

sin(2ax)         ò xmeaxdx eax å(-1)

r = 0

(m r)!ar+1

 

 

 

 

 

 

 

ò sin2

a

(Cxdx =

Definite integrals

 

2(b a)C + sin(2aC) – sin(2bC) 4C

 

L      æ np x ö     æ mp x ö

L       æ np x ö      æ mp x ö        L

 

ò sinç        ÷sinç        ÷dx = ò cosç                 ÷cosç

÷dx =

dn,m

 

0      è  L  ø     è L ø

0       è L ø

è L ø 2

 

L      æ np x ö        æ mp x ö          L2         2((-1)m+n -1) mnL2

 

ò sin ç

L ÷ x sin ç

L ÷ dx =

if m = n, or

4

(m2n2 )2p 2

if m ¹ n

 

0      è        ø        è         ø

L      æ np x ö          æ mp x ö

L3 æ         3    ö

4(-1)m+n mnL3

 

ò sin ç        ÷ x2 sin ç         ÷ dx =

ç 2 – 2 2 ÷ if m = n, or

2           2 2 2

if m ¹ n

 

0      è   L  ø          è   L ø

12 è      n p ø

(m n ) p

 

L      æ np x ö d       æ mp x ö

2mn

 

ò sinç        ÷     sinç         ÷dx = 2

 

if n m is odd, or 0 if n m is even

 

0      è   L  ø dx      è L ø

n m

 

 

 

Simple exponential integrals

 

¥

ò xneaxdx =

0

n! an+1

 

 

 

Some Gaussian integrals

 

 

 

ò x eax2  dx = 0

¥

ò xeax2  dx =  1

¥

ò eaxdx =

¥
2 a

x 2ne– ax 2 dx 1· 3· 5…(2n 1)

 

0                                    2a

¥

ò                      n n

ò                     n +1 n

¥

 

2an+1

ò x2 n+1 eax2 dx = n!

0

 

x 2ne– ax 2 dx 1· 3· 5…(2n 1)

0                                                      2    a