Time-dependent and Time-independent Schrödinger Equation for Wavefunction

Definition

The Schrödinger equation is a differential equation that governs the behavior of a quantum mechanical wavefunction $\psi$, which contains all information about the state of a particle or system.

• The time-dependent Schrödinger equation is: $$i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi$$ where $i$ is the imaginary unit, $\hbar$ is reduced Planck’s constant, and $\hat{H}$ is the Hamiltonian operator representing total energy.

• The time-independent Schrödinger equation is: $$\hat{H}\psi = E\psi$$ where $E$ is the energy eigenvalue of the system.

In simple terms, the wavefunction $\psi$ describes the quantum state, and the Schrödinger equation tells how that state behaves in space and time.

Main Content

1. Wavefunction and Physical Meaning

• The wavefunction $\psi(x,t)$ is a mathematical function that represents the state of a particle. It is not directly observable, but its square modulus $|\psi(x,t)|^2$ gives the probability density of finding the particle at position $x$ and time $t$. This probabilistic interpretation is one of the central ideas of quantum mechanics.
• A physically acceptable wavefunction must be single-valued, finite, continuous, and normalizable. Normalization means that the total probability of finding the particle somewhere in space must be equal to 1: $$\int |\psi|^2 \, d\tau = 1$$ For example, in a one-dimensional system, the probability of finding a particle between $x=a$ and $x=b$ is: $$P(a \le x \le b) = \int_a^b |\psi(x,t)|^2 dx$$

2. Time-Dependent Schrödinger Equation

• The time-dependent Schrödinger equation (TDSE) gives the complete dynamical description of a quantum system: $$i\hbar \frac{\partial \psi(x,t)}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(x,t)\right]\psi(x,t)$$ Here, the first term in the Hamiltonian represents kinetic energy and $V(x,t)$ represents potential energy. This equation applies to all quantum systems when the potential may depend on time.

• The TDSE is used to determine how a wavefunction evolves from an initial state. If the initial wavefunction is known, the TDSE can predict the future probability distribution. For example, in a free-particle case where $V=0$, the wavefunction evolves as a superposition of wave packets that spread over time.

3. Time-Independent Schrödinger Equation

• When the potential energy does not depend on time, i.e. $V(x,t)=V(x)$, the wavefunction can often be separated into spatial and temporal parts: $$\psi(x,t)=\phi(x)T(t)$$ Substituting this into the TDSE leads to the time-independent Schrödinger equation (TISE): $$\hat{H}\phi = E\phi$$ or, in one dimension, $$-\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2} + V(x)\phi = E\phi$$

• The TISE is used to find stationary states, which are states whose probability density does not change with time. These states correspond to definite energy values. A classic example is the particle in a one-dimensional box, where the boundary conditions restrict the allowed wavefunctions and produce discrete energy levels: $$E_n = \frac{n^2h^2}{8mL^2}, \quad n=1,2,3,\dots$$ This shows the concept of energy quantization in quantum mechanics.

4. Relationship Between TDSE and TISE

• The TDSE is the more general equation because it describes all quantum states, including those where the potential changes with time. The TISE is derived from the TDSE only when the potential is time-independent.
• The TISE is particularly useful for solving bound-state problems, such as electrons in atoms, molecules, and potential wells. Once the spatial part $\phi(x)$ is found, the time factor is usually: $$T(t)=e^{-iEt/\hbar}$$ So the complete stationary-state solution becomes: $$\psi(x,t)=\phi(x)e^{-iEt/\hbar}$$ The probability density for such a state is: $$|\psi(x,t)|^2 = |\phi(x)|^2$$ which is independent of time. This is why these states are called stationary.

Working / Process

1. Start with the physical system and identify the potential energy function $V(x,t)$. If the potential depends on time, use the time-dependent Schrödinger equation; if it is independent of time, separation of variables may be applied.

2. Write the Hamiltonian operator: $$\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V$$ Substitute it into the Schrödinger equation and solve the resulting differential equation using suitable boundary conditions.

3. Normalize the wavefunction and interpret the solution physically. For time-independent problems, find energy eigenvalues and eigenfunctions; for time-dependent problems, determine how the wavefunction evolves with time and calculate probabilities using $|\psi|^2$.

Advantages / Applications

• It provides a complete mathematical description of microscopic particles such as electrons, protons, and atoms.
• It is used to calculate quantized energy levels in atoms, molecules, and solids, which is essential in spectroscopy and chemical bonding.
• It explains important quantum phenomena such as tunneling, confinement, and the stability of atomic structures.

Summary

• The Schrödinger equation is the central equation of quantum mechanics for describing the wavefunction.
• The time-dependent form explains the evolution of quantum states with time, while the time-independent form helps find stationary states and energy levels.
• The wavefunction itself is interpreted probabilistically, and its modulus squared gives the probability density of finding a particle in a region of space.