Here we start to explore the wave function ( Schrödinger equation). Our hypothetical particle/wave is first placed in an infinite potential well, and we are finding it's energy, and then this followed by a finite potential well.
1 Week 2
1.1 The Wavefunction Equation
Particles behave like waves. They have a wavelength ( according to de Broglie
) λ = hp
Waves should have a wave equation, whose solutions are wave functions.
We are representing this with Ψ which is a function of space, so represented by
3D coordinate r, and time.
Ψ(r, t)
2 2
Using that E = h̄ω = h̄2m
k
we can derive the Wave Function Equation.
∂Ψ(r, t) −h̄2 2
ih̄ = ∇ Ψ(r, t) + V (r)Ψ(r, t) (1)
∂t 2m
We call V(r) the external potential. It’s conservative, which means V(r) is not
V(r,t), meaning it’s not varying in time, A wavefunction is a complex number (
due to the imaginary unit ) and it’s also NOT for photons, due to the mass in it.
1.2 The time independent wavefunction equation
Most waves can be split in to space and time variant.
Ψ(r, t) = Aei(kr−ωt) = Aeikr e−iωt (2)
So we can actually using this idea to write the following:
Ψ(r, t) = ψ(r) ∗ ϕ(t) (3)
Where ψ and ϕ are both wave functions as a wave function can be a combination
of another two.
Now, if we substitute (3) into (1) we get the following:
, Important to know, that when there is a partial with respect to time (∂t) does
not act ( operate ) on the spatial part of the wave function (ψ(r)), and similarly
∇2 does not act on the time part. (ϕ(t))
Now we can rearrange this equation to get all the time parts on one side, and
spatial parts on the other by dividing both side by ψ(r)ϕ(t):
−h̄2 2
ih̄ ∂ϕ(t)
∂t 2m ∇ ψ(r) + V (r)ψ(r)
= (6)
ϕ(t) ψ(r)
Now we have TIME only and SPACE only.
The next trick is to use the separation of variables. To make sure we have units
on each side correctly laid out, each side must be equal to a constant, which we
call separation constant ϵ.
Time part:
1 ∂ϕ(t) −iϵ
= (7)
ϕ(t) ∂t h̄
which clearly leads to
∂ϕ(t) −iϵ
= ϕ(t) (8)
∂t h̄
It has the solution:
iϵ
ϕ(t) = e−( h̄ )t (9)
This looks like a classical wave! It would have a e−iωt part which is simply the
time part of the classical wave. Therefore we can make the analogy, that h̄ϵ = ω
which is an equation for a matter wave! So the separation constant is simply
the energy of the matter wave ( E ) equals the epsilon! E = ϵ. As long as the
potential doesn’t vary in time, this is the only way the function will!
Now looking at the space part of the equation:
1 −h̄2 2
[ ∇ ψ(r) + V (r)ψ(r)] = E (10)
ψ(r) 2m
2
Therefore, [ −h̄ 2
2m ∇ + V (r)]ψ(r) = Eψ(r) is the Time-Independent Wavefunction
equation!
−h̄2 2
2m ∇ is the Kinetic Energy and V(r) is the Potential Energy.
The term in the square bracket is an operator called Hamiltonian, which we call
H. This simplifies the whole equation to
Hψ(r) = Eψ(r) (11)
Another point to make is that the Kinetic Energy is 21 mv 2 which we can write
p2
as 2m , so we can define a momentum operator:
p = −ih̄∇ (12)
2
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