Plane Wave
Consider plane wave exp[j(kx-wt)] propagating along the x-axis where w = angular frequency, t is time and k is wave number.
Those constants can be defined as:
- its temporal periodicity that equals to:
- its spatial periodicity that equals to:
The energy of this wave is given by:
(Where:
h = Planck’s constant = 6.626*10^-34 Js = 4.1357*10^-15 eVs)
The momentum equals to:
To find which operator corresponds to observable E we determine the partial derivative of the plane wave equation in terms of time, t:
Therefore:
To find which operator corresponds to observable p we determine the partial derivative of the plane wave equation in terms of position, x:
Therefore:
Now the question is which operator corresponds to observable x?
Answer:
How does x^2 work as an operator? x^2 is composite operator:
Where: xf(x) = Operator
= x(xf(x))
Where: x( ) = Operator, xf(x) = Function
= x(xf(x))
Where: x(xf(x)) = Function
How does p^2 work as an operator?
To summarise:
Going back to the classic harmonic oscillator the Hamiltonian operator is given by:
According to above derivations:
This equation is part of Schrodinger equation:
Now we can find time independent Schrodinger equation:
Let:
Thus we get time-independent Schrodinger equation:
