Schrödinger Equation: Detailed Explanation, Proofs, and Derivations
The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is named after Erwin Schrödinger, who formulated the equation in 1925.
The Time-Dependent Schrödinger Equation
The time-dependent Schrödinger equation is given by:
$$ i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \hat{H} \Psi(\mathbf{r}, t) $$
where:
– \(i\) is the imaginary unit,
– \(\hbar\) is the reduced Planck’s constant,
– \(\Psi(\mathbf{r}, t)\) is the wave function of the system,
– \(\hat{H}\) is the Hamiltonian operator,
– \(\mathbf{r}\) represents the spatial coordinates,
– \(t\) represents time.
The Hamiltonian operator \(\hat{H}\) typically includes kinetic and potential energy terms:
$$ \hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}, t) $$
where \(m\) is the mass of the particle, \(\nabla^2\) is the Laplacian operator, and \(V(\mathbf{r}, t)\) is the potential energy.
The Time-Independent Schrödinger Equation
For a system with a time-independent Hamiltonian, the Schrödinger equation can be separated into spatial and temporal parts. The time-independent Schrödinger equation is given by:
$$ \hat{H} \psi(\mathbf{r}) = E \psi(\mathbf{r}) $$
where \(\psi(\mathbf{r})\) is the spatial part of the wave function, and \(E\) is the energy eigenvalue.
Derivation of the Time-Independent Schrödinger Equation
Starting from the time-dependent Schrödinger equation:
$$ i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \hat{H} \Psi(\mathbf{r}, t) $$
we assume a solution of the form:
$$ \Psi(\mathbf{r}, t) = \psi(\mathbf{r}) \phi(t) $$
Substituting this into the time-dependent Schrödinger equation, we get:
$$ i\hbar \frac{\partial}{\partial t} (\psi(\mathbf{r}) \phi(t)) = \hat{H} (\psi(\mathbf{r}) \phi(t)) $$
which simplifies to:
$$ i\hbar \psi(\mathbf{r}) \frac{\partial \phi(t)}{\partial t} = \phi(t) \hat{H} \psi(\mathbf{r}) $$
Dividing both sides by \(\psi(\mathbf{r}) \phi(t)\):
$$ i\hbar \frac{1}{\phi(t)} \frac{\partial \phi(t)}{\partial t} = \frac{1}{\psi(\mathbf{r})} \hat{H} \psi(\mathbf{r}) $$
Since the left side depends only on \(t\) and the right side depends only on \(\mathbf{r}\), both sides must equal a constant, which we call \(E\):
$$ i\hbar \frac{1}{\phi(t)} \frac{\partial \phi(t)}{\partial t} = E $$
$$ \frac{1}{\psi(\mathbf{r})} \hat{H} \psi(\mathbf{r}) = E $$
The first equation can be solved to give:
$$ \phi(t) = e^{-iEt/\hbar} $$
The second equation is the time-independent Schrödinger equation:
$$ \hat{H} \psi(\mathbf{r}) = E \psi(\mathbf{r}) $$
Normalization of the Wave Function
The wave function must be normalized, meaning the total probability of finding the particle in all space is 1:
$$ \int_{-\infty}^{\infty} |\Psi(\mathbf{r}, t)|^2 \, d\mathbf{r} = 1 $$
For the time-independent case:
$$ \int_{-\infty}^{\infty} |\psi(\mathbf{r})|^2 \, d\mathbf{r} = 1 $$
Probability Density and Current
The probability density is given by:
$$ \rho(\mathbf{r}, t) = |\Psi(\mathbf{r}, t)|^2 $$
The probability current is given by:
$$ \mathbf{j}(\mathbf{r}, t) = \frac{\hbar}{2mi} \left( \Psi^* \nabla \Psi – \Psi \nabla \Psi^* \right) $$
These satisfy the continuity equation:
$$ \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0 $$
