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Last modified: July 29, 2022

Basic Relationships
-------------------

The Photoelectric Effect pointed to the particle 
properties of light.  DeBroglie wondered if this 
idea could be extended to other particles.  He 
started with Einstein's formula for energy:

E = mc2 = √(p2c2 + m20c4)

and PLANCK'S HYPOTHESIS:

E = hf = hω

Where h = Planck's constant.

Equating these 2 equations when m0 = 0 gives,

pc = hf

   = hc/λ

∴ p = h/λ or λ = h/p ... the DeBROGLIE WAVELENGTH

λ is associated with waves, p is associated with
particles.

Define the wavenumber, k = 2π/λ.

∴ p = hk/2π

    = hk where h is the reduced Planck's constant.

p = -ih∂/∂x ... momentum operator

E = mv2/2 

  ≡ p2/2m 

  ≡ -(h2/2m)∂2/∂x2 

  ≡ h2k2/2m

  ≡ hω

E = ih∂/∂t

|ψ(x)> = Σan|n(x)>
           n

|ψ(x,t)> = Σanexp(-iEnt/h)|n(x)>
           n

If H is not time dependent then:

|ψ(t)> = Σanexp(-iEnt/h)|n(x)>
         n

If H is time dependent then:

|ψ(t)> = Σan(t)exp(-iEnt/h)|n(x)>
         n


ih∂/∂t|ψ(t)> = H(t)|ψ(t)>

H(t) = H0 + V(t)


LHS:

ih∂ψ(t)/∂t = Σih[exp(-iEnt/h))dan(t)/dt + an(t)(-iEn/h)exp(-iEnt/h)]|n(x)>                

           = Σih[dan(t)/dt - an(t)(iEn/h)]exp(-iEnt/h)|n(x)>                
             n
RHS:

H(t)|ψ(t)> = Σ(hωn + V(t))an(t)exp(-iωnt)|n(x)>
             n
           = Σ(an(t)hωn + anV(t))exp(-iωnt)|n(x)>
             n

Equating the two:

Σih[dan(t)/dt - an(t)(iEn/h)]exp(-iEnt/h)|n(x)> = Σ(an(t)hωn + anV(t))exp(-iωnt)|n(x)>
n                                                 n

Σ[ihdan(t)/dt + an(t)hωn]exp(-iωnt)|n(x)> = Σ(an(t)hωn + an(t)V(t))exp(-iωnt)|n(x)>
n                                           n

Σih(dan(t)/dt)exp(-iωnt)|n(x)> = Σan(t)V(t)exp(-iωnt)|n(x)>
n                                n

Σ[ihdan(t)/dt - an(t)V(t)]exp(-iωnt)|n(x)> = 0
n                                            

<m(x)|Σ[ihdan(t)/dt - an(t)V(t)]exp(-iωnt)|n(x)> = 0
      n   

Σ[ih(dam(t)/dt)exp(-iωmt) - an(t)Vmn(t)exp(-iωnt) = 0
n  

ih(dam(t)/dt)exp(-iωmt) - Σan(t)Vmn(t)exp(-iωnt) = 0
                          n  

ih(dam(t)/dt) - Σan(t)Vmn(t)exp(-iωnt)exp(iωmt) = 0
                n  

ih(dam(t)/dt) - Σan(t)Vmn(t)exp(-i(ωn - ωm)t) = 0
                n

Vmn(t) = <m|V|n>

ih(dam(t)/dt) = Σan(t)Vmn(t)exp(-i(ωn - ωm)t) 
                n

dam(t)/dt = (1/ih)Σan(t)Vmn(t)exp(-i(ωn - ωm)t) 
                  n

dam(t)/dt = -(i/h)Σan(t)Vmn(t)exp(-i(ωn - ωm)t) 
                  n
.
am(t) = -(i/h)Σan(t)Vmn(t)exp(-i(ωn - ωm)t) 
              n

Schrodinger Wave Equation
-------------------------

Assume a free particles can be represented as a traveling 
wave defined by:

ψ(x,t) = Aexp(i(kx-ωt)) 

Spatial derivative:

∂ψ/∂x = ikψ = (ip/h)ψ 

Multiply both sides by -ih to get:

-ih∂ψ/∂x = pψ  

Square and divide by 2m to get:

-(h2/2m)∂2ψ/∂x2 = (p2/2m)ψ 

                = Eψ   ... 1.
    
This is the TIME INDEPENDENT form.
                                        
Time derivative:

In addition to its role in determining system energies, the 
Hamiltonian operator also generates the time evolution of 
the wavefunction. 

∂ψ/∂t = -iωψ = (-iE/h)ψ since E = hω

Multiply both sides by ih to get:

∴ ih∂ψ/∂t = Eψ  ... 2.

Equating 1. with 2. we get:
                                                      
-(h2/2m)∂2ψ/∂x2 = ih∂ψ/∂t 

Now p = -ih∂/∂x so the LHS becomes:

p2/2m = ih∂ψ/∂t 

or,

ih∂ψ/∂t = Eψ

This is the TIME DEPENDENT form.

The time-independent Schrodinger equation is the equation 
describing stationary states.  It is only used when the 
Hamiltonian itself is not dependent on time.  Fortunately, 
the majority of interesting problems in quantum mechanics 
involve stationary states with definite total energy and 
so the time dependent form is not required.

Equation in the presence of a potential:

Time Independent form:

-(h2/2m)∂2ψ(x)/∂x2 + U(x)ψ(x) = Eψ(x)

Time Dependent form:

-(h2/2m)∂2ψ(x,t)/∂x2 + U(x)ψ(x,t) = ih∂ψ(x,t)/∂t

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