Download Review Material for Physics 250 and more Study notes Quantum Mechanics in PDF only on Docsity!
REVIEW MATERIAL FOR PHYSICS 250
- Lorentz transformation:
- addition of velocities
- Momentum and energy: , , ,
- Doppler Shift: ,
- Statistical Physics
Maxwell-Boltzmann distribution
, in 3 D
S ' moving along + x axis
x =γ ( x ′ + c β t ′) ct = γ ( ct ′ +β x ′) x^2 – c^2 t^2 = x ′^2 – c^2 t ′^2
u (^) x
u ' x + v 1 +( u ′ x v ⁄ c^2 )
u (^) y^1 γ
u ′ y 1 +( ( u ′ x v ) ⁄ c^2 )
p ˜^ = ( E ⁄ c , p ) E = γ m (^) o c^2 p = βγ m (^) o c β pc E
E^2 = p^2 c^2 + m (^) o^2 c^4
E^2 – p^2 c^2 = E ′^2 – p ′^2 c^2 E = γ ( E ′ +β p ′ x c ) px c = γ ( p ′ x c +β E ′)
mv = qBR
pc ( GeV) = 0.3 q B ( Tesla) R m ( )
f^1 +β 1 – β
------------^
1 ⁄ 2 = f ′ λ 1 – β 1 +β
------------^
1 ⁄ 2 = λ′
n ( E ) dE g E ( ) f ( E ) dE^2 π N ( π kT )^3 ⁄^2
= =^ -----------------------^ E e – E^^ ⁄ kTdE
n ( ) v dv 4 π N v^2 m 2 π k (^) B T
----------------^
3 ⁄ 2 e
1 2 --- mv
-^2 ⁄ k (^) B T
〈 K 〉 1
〈 --- mv^2 〉 3 2
= = --- k (^) B T v rms
3 k (^) B T m
〈 --- mv^2 x 〉 1 2 = --- k (^) B T
- Early Quantum Physics
- Stefan-Boltzmann Law:
- The Photelectric effect:
Particle Properties of Waves
(a) Compton scattering:
(b) Absorption of Photons:
(c) Gravitational Red Shift:
Rutherford Scattering;
〈 ε ν( )〉 h ν e
h ν ⁄ k (^) BT
u ( ν) 8 π h ν
3
c^3
e
h ν ⁄ k (^) B T
hc λ m k (^) B T
R = σ T^4
σ =5.67∗ 10 –^8 W ⁄ m^2 K^4
h ν =φ + T (^) e^ max
λ – λ 0 = mc^ ------^ h - 1( – cosθ)
N ( x ) = N 0 e – μ x
ν 2 ν 1 1 gL C^2
ν' ν 1 GM Rc^2
N ( θ) k
(^2) Z (^2) e (^4) Nnt
4 r^2 T (^) α^2 sin^4 ( θ ⁄ 2 )
N = # alphas/ m^2 n = # atoms/ m^3 in foil t = thickness of foil
b kZ e
2 T (^) α = ------------^ cotθ ⁄ 2 r = distance of detector from foil
Tunneling:
Schrodinger Equation in higher dimensions:
- Operators and Expectation Values
Two dimensional box:
Central Forces:
Spherical harmonics and total angular momentum
,
Probabilities:
Expectation values:
Spherically symmetric potential:
T E ( ) 16 VE
0
------ 1 E
V 0
– ------^ e^
- 2 k 2 L ≈ k 2 2 m V (^^ – E ) h^2
p h i
∂ x
∂ y
∂ z
= (^) , , ----- p^2 – h^2 ∂
2
∂ x^2
2
∂ y^2
2
∂ z^2
= =– h^2 ∇^2
Lz x p (^) y y p (^) z h i
∂φ
E h
2 2 m
2 π n (^) x L (^) x
------------^
(^2 2) π n (^) y L (^) y
------------^
2
2 8 m
n (^) x L (^) x
-----^
(^2) n (^) y L (^) y
-----^
2 = = +
ψ ( x y , ) 2 L (^) x
L (^) y
π n (^) x x L (^) x
π n (^) y y L (^) y
= sin sin ------------
P x ( 1 < x < x 2 , y 1 < y < y 2 ) dx dy ψ ( x y , ) 2 y 1
y 2
x 1 ∫
x 2
Y (^) lm ( θ φ, ) = Θ lm ( θ) e im φ m = 0 ,± 1 ,± 2 ,… ,± l
L^2 Y (^) lm ( θ φ, ) = h^2 l l ( + 1 ) Y (^) lm (θ φ , ) L (^) z Y (^) lm ( θ φ, ) = hm (^) l Y (^) lm ( θ φ, )
P (θ 1 < θ < θ 2 ,φ 1 < φ <φ 2 ) d θ θ d φ Y (^) lm ( θ φ, ) 2 φ 1
φ 2
θ 1 sin^ ∫
θ 2
〈 f ( θ φ, )〉 =∫sin θ( d θ) φ d f ( θ φ, ) Y lm (θ φ , ) 2
U ( ) r = U r ( )
ψ nlm ( ) r = R (^) nl ( ) rY (^) lm ( θ φ, )
Radial probabilities and averages:
Hydrogenic atoms:
Magnetic moments:
Orbital:
Electron Spin S :
, , ,
,
Protons and Neutrons:
, , ,
Electron and orbital spin:
Fermi Sea: , ; , ,
∇^2 [ R r ( ) Y (^) lm ( θ φ, )] ∂
2
∂ r^2
r
∂ r
----- l l (^^ +^1 ) r^2
= + – ----------------- R r ( ) Y (^) lm ( θ φ, )
h^2 2 m
2 R
dr^2
r
--- dR dr
2
2 mr^2
- ----------------------- R r ( ) + U r ( ) R r ( ) = ER r ( )
P r ( 1 < r < r 2 ) r^2 dr R r ( ) 2 r 1
r 2
〈 f ( ) r 〉 drr^2 f ( ) r R r ( ) 2 0
∞
U r ( ) kZ e
2 r = –^ --------- E (^) n k e
2 2 a 0
-------- Z
2
n^2
μ B =5.788 × 10 – 5 eV/T
E = – μ ⋅ B μ = – μ B ( L/ h ) E = μ B m (^) l B
S^2 = s s ( + 1 ) h^2 S (^) z = m (^) s h s = 1 ⁄ 2 m (^) s =± 1 ⁄ 2
μ = – g μ B ( S ⁄ h ) =– 2 μ B ( S ⁄ h ) E = 2 μ B m (^) s B
μ n = eh ⁄ ( 2 m (^) p )=3.152 × 10 – 8 eV/T and s = 1 ⁄ 2
μ p n , = 2 g (^) p , n μ n ( S ⁄ h ) g (^) p = 2.79 g (^) n = – 1.91 ∆ E = 2 g (^) p n , μ n B
E = ( m (^) l + 2 m (^) s )μ B B
k (^) F 3 π^2 N V
----^
1 ⁄ 3 = 3 D k (^) F^ π 2
--- N
L
= ---- 1 D E (^) f h
(^2) k 2 2 m
n ( E ) dE
3 N
------- E – f^3 ⁄^2 E dE
e
( E – E (^) f ) ⁄ ( kT )
