Classical Action

One-dimensional system

Action
S[x(t)] = ∫(L[x,x^*,t]*dt) t_a to t_b

Lagrangian
L[x,x^*,t] = T - V = mv²/2 - V(x)

In classsical mechanics, the solution x(t) is the path x^-(t) that will give the minimum action. This condition gives the Euler-Lagrange equation:

d/dt{∂L/∂x^*)} - ∂L/∂x = 0

Example: Free particle
Calculate the action of the path of a free particle in one dimension.
L = m*x^*2/2

Solution
1. ∂L/∂x = 0
2. ∂L/∂x^*) = m*x^*
3. d/dt{result in 2} = m*x^**
4. Euler-Lagrange equation is m*x^**= 0
5. E-L equation gives x(t) = A + B*t
6. Let x_a = x(t_a) and x_b = x(t_b)
7. x^* = B
8. From #5 and #6, B = (x_b-x_a)/(t_b-t_a)
9. L = m*(B)²/2 = m*B²/2
10. S = ∫(m*B^2/2 * dt) = m*B^2/2*∫(dt) from t_a to t_b
Finally,
S = m*(x_b-x_a)^2/(t_b-t_a)

That's all!

Schrodinger equation (1D)

discussed last saturday july 9 for maricris.

Time-independent potential
The SE is just an eigenvalue problem for the hamiltonian H = T+V. That is,

Hφ(x) = Eφ(x)

where
T = kinetic energy = (1/2)mv² = p²/2m
V = V(x) <--- position representation

In the position representation, the momentum p is given by

p^→ = (hbar/i)Del^→

or in 1D

p_x = (hbar/i)d/dx

Thus, 1D SE in position representation becomes

-(hbar²/2m)d²φ/dx² + V(x)φ(x) = Eφ(x)

Complete SE (time-dependent)

HΨ(x,t) = i*hbar*∂Ψ(x,t)/∂t

To arrive at the time-independent case, just remember the following position-time-separated form of Ψ(x,t):

Ψ(x,t) = e^-i*E*t/hbarφ(x)

That's all.

For comments, suggestions, and reactions, email at blog-quantum@pisika.org.

Chapters

no need to introduce this site. let the contents speak for themselves. hehe

Chapter 1: The fundamental concepts of quantum mechanics
Chapter 2: The quantum-mechanical law of motion
Chapter 3: Developing the concepts with special examples
Chapter 4: The Schrodinger description of quantum mechanics
Chapter 5: Measurements and operators
Chapter 6: The perturbation method in quantum mechanics
Chapter 7: Transition elements
Chapter 8: Harmonic oscillators
Chapter 9: Quantum electrodynamics
Chapter 10: Statistical mechanics
Chapter 11: The variational method
Chapter 12: Other problems in probability
Appendix: Some useful definite integrals
Index