3.1: Review - Energy Eigenvalue Problem
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The time-independent wavefunction obeys the time-independent Schrödinger equation:
\[\boxed{\mathcal{H} \varphi(\vec{x})=E \varphi(\vec{x})} \nonumber\]
where E is identified as the energy of the system. If the wavefunction is given by just its time-independent part, \( \psi(\vec{x}, t)=\varphi(\vec{x})\), the state is stationary. Thus, the time-independent Schrödinger equation allows us to find stationary states of the system, given a certain Hamiltonian.
Notice that the time-independent Schrödinger equation is nothing else than the eigenvalue equation for the Hamiltonian operator.
The energy of a particle has contributions from the kinetic energy as well as the potential energy:
\[\mathcal{H}=\frac{1}{2 m}\left(\hat{p}_{x}^{2}+\hat{p}_{y}^{2}+\hat{p}_{z}^{2}\right)+V(\hat{x}, \hat{y}, \hat{z}) \nonumber\]
or more explicitly:
\[\mathcal{H}=-\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}\right)+V(x, y, z) \nonumber\]
which can be written in a compact form as
\[\boxed{\mathcal{H}=-\frac{\hbar^{2}}{2 m} \nabla^{2}+V(x, y, z)} \nonumber\]
(Notice that V (x, y, z) is just a multiplicative operator, in the same way as the position is).
In 1D, for a free particle there is no potential energy, but only kinetic energy that we can rewrite as:
\[\mathcal{H}=\frac{1}{2 m} p^{2}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}\nonumber\]
The eigenvalue problem \( \mathcal{H} w_{n}(x)=E_{n} w_{n}(x)\) is then the differential equation
\[\mathcal{H} w_{n}(x)=E_{n} w_{n}(x) \rightarrow-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} w_{n}(x)}{\partial x^{2}}=E_{n} w_{n}(x) \nonumber\]
For a free particle there is no restriction on the possible energies, En can be any positive number. The solution to the eigenvalue problem is then the eigenfunction:
\[w_{n}(x)=A \sin \left(k_{n} x\right)+B \cos \left(k_{n} x\right)=A^{\prime} e^{i k_{n} x}+B^{\prime} e^{-i k_{n} x} \nonumber\]
which represents two waves traveling in opposite directions.
We see that there are two independent functions for each eigenvalue En. Also there are two distinct momentum eigenvalues \(\pm k_{n}\) for each energy eigenvalue, which correspond to two different directions of propagation of the wave function \(e^{\pm i k_{n} x}\).