Recall the Schrödinger equations:
The first is the time-independent equation which uses the eigenfuctions for the energy operator (Hamiltonian) and the second is the time-dependent state vector equation.
Now suppose that we have a specific energy potential ; say . Note: in classical mechanics this follows from Hooke’s law: . In classical mechanics this leads to the following differential equation: which leads to which has general solution where The energy of the system is given by where is the maximum value of which, of course, is determined by the initial conditions (velocity and displacement at ).
Note that there are no a priori restrictions on . Notation note: stands for a real number here, not an operator as it has previously.
So what happens in the quantum world? We can look at the stationary states associated with this operator; that means turning to the first Schrödinger equation and substituting (note ):
Now let’s do a little algebra to make things easier to see: divide by the leading coefficient and move the right hand side of the equation to the left side to obtain:
Now let’s do a change of variable: let Now we can use the chain rule to calculate: . Substitution into our equation in and multiplication on both sides by yields:
Since is just a real valued constant, we can choose .
This means that
So our differential equation has been transformed to:
We are now going to attempt to solve the eigenvalue problem, which means that we will seek values for that yield solutions to this differential equation; a solution to the differential equation with a set eigenvalue will be an eigenvector.
If we were starting from scratch, this would require quite a bit of effort. But since we have some ready made functions in our toolbox, we note 🙂 that setting gives us:
This is the famous Hermite differential equation.
One can use techniques of ordinary differential equations (say, series techniques) to solve this for various values of .
It turns out that the solutions are:
Graphs of the eigenvectors (in ) are here:
Of importance is the fact that the allowed eigenvalues are all that can be observed by a measurement and that these form a discrete set.
Ok, what about other operators? We will study both the position and the momentum operators, but these deserve their own post as this is where the fun begins! 🙂