The Schrodinger Equation#

The Schrodinger equation is an equation followed by all of non-relativstic quantum mechanics. It states that for a quantum mechanical system there exist a wavefunction \(\Psi(x,t)\) that satisfies the Schrodinger equation given by,

\[ \boxed{i\hbar \frac{\partial \Psi}{\partial t} = \hat{H} \Psi} \]

  • \(\Psi\) : The wavefunction generally written as \(\Psi(x,t)\)

  • \(\hat H\) : The Hamiltonian of the system consist of operators that describes the energy of the system.

The Schrodinger equation is often describe as the equivalent of Newton’s second law (\(F=ma\)) for quantum mechanics. Recall that Newton’s second law is a partial derivative,

\[ \begin{align} -\frac{\partial V}{\partial x} &= m\frac{d^2x}{dt^2} \tag{Partial Derivative Form} \end{align} \]

Separation By Variables#

The separation by variables may be applied to the Schrodinger Equation to separate into the product of two solutions given by,

\[\boxed{\Psi(x,t) = \psi(x)\varphi(t), \quad \varphi(t) = e^{\frac{-iEt}{\hbar}}}\]

  • \(\psi\) : the spacial wavefunction defined as \(\psi(x) \equiv \Psi(x)\)

  • \(\varphi\) : the temporal wavefunction defined as \(\varphi(t) \equiv \Psi(t)\).

If you were to plug in the separated solution into the Schrodinger equation you will get a very important relationship,

\[ \frac{i\hbar }{\varphi}\frac{\partial \varphi}{\partial t} = H\hat \psi = \text{constant} \]

Derivation#

Time Independent Schrodinger Equation#

The Time Independent Schrodinger Equation (TISE) is the separated spacial equation of the Schrodinger Equation for the spacial wavefunction \(\psi(x)\) given by,

\[ \boxed{\hat H \psi = E\psi} \]

  • E : Total energy of the particle.

  • The TISE is special equation called the eigenvector \(\psi\) and eigenvalue \(E\) problem. It states that for some system describe by its Hamiltonian there exist solution(s) (the wavefunction) with energy \(E\).

  • If there are more than one solution with energy \(E\) we consider those solution to be degenerate.

General Solution#

Once again we would love to know the solution (a wavefunction) of a system given by some Hamiltonian \(\hat H\). We cannot do so without solving for the Time Independent Schrodinger Equation which gives us the spacial wavefunction \(\psi(x)\). Knowing \(\psi(x)\) we can write a general solution,

\[ \Psi(x,t) = \sum{c_n\Psi_n(x,t)} = \sum{c_n\psi_n(x)\varphi_n(t)}\]
\[ \boxed{\Psi(x,t) = \sum{c_n\psi_ne^{\frac{-iE_nt}{\hbar}}}} \]

Examples#

Infinite Square Well#

Finite Square Well (Scattering State)#

Finite Square Well (Bounded State)#

Free Particle#

Potential Step (Scattering State)#

Potential Step (Bound State)#

Delta Potential (Scattering State)#

Delta Potential (Bound State)#