2.3. The Hamiltonian and Schrodinger’s Wave Equation#

In the last section we showed that to properties such as momentum can be extracted from the wave function using the appropriate operator. In most systems the particle will have a kinetic energy and be subject to a potential. For particles confined to one dimension we already know the operators for both of these quantities. Combined these operators are called the Hamiltonian and tell us about the energy of the particle.

2.3.1. The Hamiltonian#

The Hamiltonian is the operator that describes the energy of a particle. This is the sum of the kinetic and potential energy operators.

(2.9)#\[\begin{split} \hat{H}&=-\hat{E}_k+\hat{V}(x)\\ &=-\frac{\hbar ^2}{2m}\frac{d^2}{dx^2}+V(x) \end{split}\]

where \(-\frac{\hbar ^2}{2m}\frac{d^2}{dx^2}\) is the kinetic energy operator and \(V(x)\) is the potential energy operator, both in just the x-dimension. We can also obtain the energy of the particle by using the operator \(\hat{E}=i\hbar\frac{d}{dt}\).

2.3.2. The Schrodinger wave equation#

By letting both the Hamiltonian and the Energy operator act on the wave function we define the behaviour of the particle.

(2.10)#\[\begin{split} \hat{H}\Psi(x,t)&=\hat{E}\Psi(x,t)\\ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\Psi(x,t)+V(x)\Psi(x,t)&=i\hbar\frac{d}{dt}\Psi(x,t) \end{split}\]

This is the time-dependent Schrodinger wave equation. In many of our systems the probability density will be constant with time because we can express the wave function \(\Psi(x,t)=\psi(x)e^{-i\omega t}\). For such systems we can simplify the Schrodinger equation to it’s time independent form.

(2.11)#\[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x)=E\psi(x) \]

Once again, this is the kinetic energy operator \(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\) and the potential energy operator \(V(x)\) acting on the wave function. The solutions are the energy eigenvalues \(E\) for the wave function \(\psi(x)\). We will states whose wave functions are solutions to the time independent Schrodinger equation Stationary States due to the lack of time dependence for the probability density function.