This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems — the particle in a box, the harmonic oscillator, the rigid rotor and the ...
• Nearly any system near equilibrium can be approximated as a H.O.
• One of a handful of problems that can be solved exactly in quantum
mechanics
examples
m1 m2 B (magnetic
field)
A diatomic molecule µ
(spin
magnetic
moment)
E (electric
field)
Classical H.O.
m
k
X0 X
Hooke’s Law: (
f = −k X − X 0 ≡ −kx)
(restoring force)
d2x d2x ⎛ k ⎞
f = ma = m 2 = −kx ⇒ + x=0
dt dt 2 ⎜⎝ m⎠⎟
,5.61 Fall 2007 Lectures #12-15 page 2
Solve diff. eq.: General solutions are sin and cos functions
k
() ( )
x t = Asin ω t + B cos ω t ( ) ω=
m
or can also write as
() (
x t = C sin ω t + φ )
where A and B or C and φ are determined by the initial conditions.
e.g. ()
x 0 = x0 ()
v 0 =0
spring is stretched to position x 0 and released at time t = 0.
Then
() ()
x 0 = A sin 0 + B cos 0 = x0 () ⇒ B = x0
dx
()
v 0 =
dt
()
= ω cos 0 − ω sin 0 = 0() ⇒ A=0
x=0
So ()
x t = x0 cos ω t ( )
k
Mass and spring oscillate with frequency: ω =
m
and maximum displacement x0 from equilibrium when cos(ωt)= ±1
Energy of H.O.
Kinetic energy ≡ K
2
1 1 ⎛ dx ⎞ 1 1
( ) ( )
2
K = mv 2 = m ⎜ ⎟ = m ⎡⎣ −ω x0 sin ω t ⎤⎦ = kx02 sin 2 ω t
2 2 ⎝ dt ⎠ 2 2
Potential energy ≡ U
dU 1 1 2
()
f x =−
dx
⇒ U = − ∫ f x dx =() ∫ ( kx )dx =
2
kx 2
=
2
( )
kx0 cos 2 ω t
, 5.61 Fall 2007 Lectures #12-15 page 3
Total energy = K + U = E
1 2 1 2
E=
2
( ) ( )
kx0 ⎡⎣sin 2 ω t + cos 2 ω t ⎤⎦ E= kx
2 0
x (t )
x 0(t )
0 t
-x0(t)
U K
1 2
kx E
2 0
0 t
Most real systems near equilibrium can be approximated as H.O.
e.g. Diatomic molecular bond A B
X
U
X
X0 A + B separated atoms
equilibrium bond length
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