The slides are designed for my guided study in MSc CUHK. It is about the brief description on classical mechanics and quantum mechanics . Some Slides I got from the slideshare clipboards for better illustration of the ideas in Physics. Thanks to slideshare, I make a milestone on presenting one of the prominent fields in modern physics.

1. Guided Study: A Quick Run
Through Classical and Quantum
Mechanics
Msc Student: Alan Leung Shek Lun
Advisor: Professor Hui Pak Ming

2. Outline
Classical Mechanics (mass-spring system)
1. Newton’s laws of motion
2. Momentum
3. Work-energy Theorem
4. Potential Energy and force
(relation)
5. Harmonic Oscillator
6. Lagrangian Mechanics and
Hamiltonian Mechanics
Quantum Mechanics
1. Schrodinger Equation
2. Typical 1D problems
3. Typical 2D and 3D problems
and their key features
4. Quantum Harmonic Oscillator

3. Classical Mechanics
mass-spring system

4. Newton’s law of motion
At time t=0s, the box is in equilibrium,
i.e. Fnet = 0N
Displace a distance xm for oscillation
The force always acts toward the equilibrium is restoring force.
By Hook’s law,
F = -kx
F µ x
Negative sign:pointing in opposite direction

5. Newton's law of motion
The motion of the box is periodic,
like a sinusoidal function,
we can be express x(t) = Asin(wt +f),
Diiferentiate both sides with respect to t,
v(t) = x'(t) = wAcos(wt +f), vmax = wA
a(t) = x''(t) = -w2
Asin(wt +f), amax = w2
A
A is the boundary condition, x £ A,
since it is the internal force that drives the motion.

6. Newton’s law of motion
m(-Aw 2
sin(wt +f))+ kAsin(wt +f) = 0
k
m
A = w2
A,
w =
k
m
= 2p f
Also, it is a form of simple harmonic motion:
d2
x
dt2
= -
k
m
x = -w2
x
By Newton's law of motion
F = ma
-kx = m
d2
x
dt2
m
d2
x
dt2
+ kx = 0,which is a second order differential equation
The solution to this differential equation is of the form:
x(t) = Asin(wt +f)

7. Relation of Force and Potential Energy
For some conservative forces,
Slope = F = -
dV(x)
dx
= -kx
At equilibrium point, F =
dV(x)
dx
= 0N
When
dV(x)
dx
> 0, F < 0, vice versa.
Integrating both sides with respect to x and get,
Area = Fò dx = -dV(x)ò = -kxdxò
V(x) = - -dV(x) = - -kxdxòò =
kx2
2
V(x) µ k
The narrowness of the curve depends k.

8. Work and Energy Theorem
F =
d
dt
(mv)
F dx =
1
2
ò mvdx
1
2
ò
= m dv
dx
dt1
2
ò
=
1
2
m(v2
2
- v1
2
)
= T2 - T1
= KE
F = -
dV(x)
dx
F dx
1
2
ò = - dV(x)
1
2
ò
= -V2 +V1
= PE

9. Total Mechanical Energy
PE = KE
-V2 +V1 = T2 -T1
T1 +V1 = T2 +V2
Total mechanical energy = Kinetic Energy + Potential Energy

10. Conservation of Energy
E = T +V
=
1
2
mv2
+
1
2
kx2
=
1
2
m(Aw cos(wt +f))2
+
1
2
k(Asin(wt +f))2
=
1
2
kA2
(cos(wt +f)2
+ sin(wt +f)2
) (∵ w2
=
k
m
)
=
1
2
kA2
, which is a constant.
Energy is conserved.

11. Boundary Condition
When the box is at its maximum point,
PE = max, KE = 0J
E = PE
=
1
2
kA2
Total energy is constant, i.e.
harmonic oscillator is a conservative system.
When the box is at its equlibrium point,
PE = 0J, KE = max
E = KE
=
1
2
mv2
max

12. Phase Diagram of momentum and position
E = T +V
E =
1
2
mv2
+
1
2
kx2
1
2
kA2
=
px
2
2m
+
1
2
kx2
(∵ p = mv)
1=
px
2
mkA2
+
x2
A2
This is an equation of ellipse,
px
2
b2
+
x2
a2
= 1, a = A2
& b = mkA2
When p = 0,
1
2
kA2
=
1
2
kx2
Þ x = A
When x = 0,
1
2
kA2
=
px
2
2m
Þ px = mkA2

13. Lagrangian Mechanics

14. Euler-Lagrange Equation

17. Conservative law

18. Conservative law
If the transformation correspond to a symmetry transformation,
Lagrangian is invariant, d L = 0
d
dt
pidqi( )
i
å = 0
pidqi( )
i
å = pi fi (qi )( )
i
å = Q
which is a certain quantity, so momentum is conserved.

19. In Lagrangian, we have talked about
least action of motion, momentum
where is ENERGY?

20. Change in coordinate
dq ®dq +d
If d L = 0, L is invariant, the sysmmetric
There is conservative law.

21. Hamiltonian Mechanics

22. Conservation of Energy

23. Total Mechanical Energy

26. 5 postulates in Quantum
Mechanics

31. Quantum Mechanics
Schrodinger Equation

34. For harmonic oscillator, it can be expressed as

36. Quantum Mechanics
Typical 1D problem

37. A free particle

39. Particle in an infinite well

40. Finite Potential Well (bound states)

43. Simulation
The one dimensi onal particle in a Box

44. Simulation
Superposition States in an infinite square well

45. Quantum Mechanics
2D and 3D problems and their key features

46. Simulation
Energy Eigenfunction of the two dimensional infinite well

48. Quantum Mechanics
Harmonic Oscillator

49. Ladder Operator Method

50. Quantum Harmonic Oscillator Energy level

51. When the energy is getting larger,
It is more close to Classical phonomenen