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Quantum mechanics formula sheet, Cheat Sheet of Quantum Mechanics

Colby CollegeQuantum Mechanics

Typology: Cheat Sheet

2021/2022
On special offer
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Quantum Mechanics Formula Sheet
Ephoton = h= 1
2 mv2 + e  1
2 mv2 +  E = h= –h
1
n22 – 1
n12
eix = cos x + i sin x ei + e-i
2 = cos  ei – e-i
2i = sin  eikx k = p
ħ = (2mE)½
ħ
E= n2ħ22
2ma2= n2h2
8ma2 =
2
a
½sinnx
a (x,y)=
4
ab
½sinnxx
a sinnyy
b E= h2
8m
n12
a2+n22
b2
xx
^ pħ
i
x tt
^ E(t) E
^ = iħ
t Ek – ħ2
2m 2
x2 [O1,O2] = O
^1O
^2 – O
^2 O
^1
(x) = A e–½ a2x2 2 = mk
ħ A =
2
¼ (x) =
mo
ħ
¼e–mox2
2ħ
x = r sincos y = r sinsin z = r cos d = r2 sindr dd
r2 = x2 + y2 + z2 : 0   : 0  2r: 0 r
L
^x = – iħ
y
z – z
y L
^y = –iħ
z
x – x
z L
^z = –iħ
x
y – y
x
2 = 1
r
2
r2r +
1
r222 = 1
sin2
2
2+
1
sin
sin
|L|= ħ l(l+1)
Lz = ħml =
1
2
1/2eiml – ħ2 2Yl,ml l (l + 1)h
–2 Yl,ml E = ħ2
2I l ( l + 1)
– ħ2
2m
1
r d2
dr2 r R – Z e2
4or RR En = –
Z2 e4 m
322o2 ħ2 1
n2 ao = 4o ħ2
me2
En = – Z2 ħ2
2mao2 1
n2 En = –13.6 eV Z2
n2 = –109,678 cm-1 Z2
n2 = –1312 kJ mol-1 Z2
n2 = – H
2 Z2
n2
1s = 1
Z
ao
3/2e-Zr/ao h = 6.626 x 10-34 J s ħ = 1.055 x 10-34 J s
2s = 1
4 2
Z
ao
3/2
2 – Zr
ao e-Zr/2ao e = 1.602 x 10-19 C
2pz = 1
4 2
Z
ao
3/2 e-Zr/2ao Zr
ao cos o = 8.85 x 10-12 J-1c2m-1
211 = 1
4 2
Z
ao
3/2 e-Zr/2ao Zr
ao sinei ao = 0.0529 nm = 0.529Å
21-1 = 1
4 2
Z
ao
3/2 e-Zr/2ao Zr
ao sine-i 1H = 2625.5 kJ mol-1
2px = 211 + 21-1
2 = 1
4 2
Z
ao
3/2e-Zr/2ao Zr
ao sincos  me = 9.11x10-31 kg
2py = 211 – 21-1
2i = 1
4 2
Z
ao
3/2e-Zr/2ao Zr
ao sinsin  h = 109,678 cm-1
pf2
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Quantum Mechanics Formula Sheet

Ephoton = h=

mv

2

  • e  

mv

2

  •   E = h= –h

n 2

2

n 1

2

e

ix = cos x + i sin x

e

i

  • e
  • i

= cos 

e

i

  • e
    • i

2i

= sin  e

ikx k =

p

ħ

(2mE)

½

ħ

E=

n

2 ħ

2 

2

2ma

2 =^

n

2 h

2

8ma

2

a

½

sin

nx

a

(x,y)=

ab

½

sin

nxx

a

sin

nyy

b

E=

h

2

8m (^) 

n 1

2

a

n 2

2

b

2

xx

^ p

ħ

i 

x

tt

^

E(t)  E

^

= iħ

t

Ek 

  • ħ

2

2m

2

x

2

[O 1 ,O 2 ] = O

^

1 O

^

2 – O

^

2 O

^

1

(x) = A e

–½ a

2 x

2

2

mk

ħ

A =

2

¼

(x) =

mo 

ħ

¼

e

mox

2

2ħ

x = r sincos y = r sinsin z = r cos d = r

2 sindr dd

r

2 = x

2

  • y

2

  • z

2 : 0    : 0   2 r: 0  r  

L

^

x = –^ iħ^

y

z

  • z

y

L

^

y = –iħ^

z

x

  • x

z

L

^

z = –iħ^

x

y

  • y

x

2

r 

2

r

2

r +

r

2

2

sin

2  (^) 

2

2

sin 

sin

|L|= ħ l(l+1)

Lz = ħm l

1 / 2

e

iml

  • ħ

2 

2 Y l,

m l

 l (l + 1)h

Y

l

,m l

E =

ħ

2

2I

l ( l + 1)

  • ħ

2

2m 

r

d

2

dr

2

r R –

Z e

2

4 or

RR En = –

Z

2 e

4 m

2 o

2 ħ

2

n

2 ao =^

4 o ħ

2

me

2

E

n

Z

2 ħ

2

2mao

2

n

2

E

n

= –13.6 eV

Z

2

n

2

= –109,678 cm

Z

2

n

2

= –1312 kJ mol

Z

2

n

2

H

Z

2

n

2

1s =

Z

ao

3/

e

-Zr/ao h = 6.626 x 10

  • J s ħ = 1.055 x 10 - J s

2s =

Z

ao

3/

Zr

ao

e

-Zr/2ao e = 1.602 x 10

  • C

2pz =

Z

ao

3/

e

-Zr/2ao

Zr

ao

cos o = 8.85 x 10

  • J - c

2 m

Z 

ao

3/

e

-Zr/2ao

Zr

ao

sine

i ao = 0.0529 nm = 0.529Å

Z

ao

3/

e

-Zr/2ao

Zr

ao

sine

  • i    1H = 2625.5 kJ mol

2px =

Z

ao

3/

e

-Zr/2ao

Zr

ao

sincos  me = 9.11x

  • kg

2py =

2i

Z

ao

3/

e

-Zr/2ao

Zr

ao

sinsin   h

= 109,678 cm

l ml Y

l,m l

0 0 (1/4π)

½

(3/4π)

½ cos

±(3/8π)

½ sine

±i

(^2 0) (5/16π)

½ (3 cos

2  – 1)

±(15/8π)

½ cossine

±i

±(15/32π)

½ sin

2 e

±i2

y =  x  

(y) = H e

–½ y

2 d

2 H 

dy

2 – 2y^

dH 

dy

+ 2 H

 H(y)^ H 

(x)

1 2y (^2) x

2 4y

2

  • 2 4 

2 x

2

  • 2

3 8y

3

  • 12y (^8) 

3 x

3

  • 12x

H

+

= 2y H 

- 2 H

-

-∞

∞

H

'

e

  • ½ y

2

H

e

  • ½ y

2

dy = 0 if ' 

= π

½ 2

 ! if ' 

sin

2 (x) dx = – ½ sin(x) cos(x) + ½ x 

cos

2 (x) dx = ½ sin(x) cos(x) + ½ x

sin(x) cos(x) dx = ½ sin

2 (x)

0

/

sin

2 (x) dx = 

0

/

cos

2 (x) dx =

0

/

sin

3 (x) dx = 

0

/

cos

3 (x) dx =

0

sin

2 (ax) dx = 

0

cos

2 (ax) dx =

0

/a

cos(ax) sin(ax) dx = 

0

cos(ax) sin(ax) dx = 0

0

sin(ax) sin(bx) dx = 

0

cos(ax) cos(bx) dx = 0 (a  b; a,b integers)

0

sin(ax) cos(bx) dx =

2a

a

2 -b

2 if a-b is odd, or zero if a-b is even

0

cos(ax) sin(ax) dx = 0 

0

n

x

2 sin

2 (x) dx =

n

3 

3

n