Schrödinger Wave Equation

The Schrödinger wave equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is the quantum mechanical analog of Newton's second law of motion in classical mechanics. There are two forms of the Schrödinger equation: the time-dependent form and the time-independent form.

Time-Dependent Schrödinger Equation (TDSE)

The TDSE describes the evolution of the wave function $\Psi(x, t)$ of a quantum system over time. For a single particle of mass $m$ moving in a potential $V(x, t)$, the equation is:

$i\hbar \frac{\partial}{\partial t} \Psi(x, t) = \left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x, t) \right) \Psi(x, t)$

Where:

• $i$ is the imaginary unit ($\sqrt{-1}$)
• $\hbar$ is the reduced Planck constant ($\hbar = h/2\pi$)
• $\frac{\partial}{\partial t}$ is the partial derivative with respect to time
• $\frac{\partial^2}{\partial x^2}$ is the second partial derivative with respect to position
• $m$ is the mass of the particle
• $V(x, t)$ is the potential energy of the particle
• $\Psi(x, t)$ is the wave function of the particle

The term in the parenthesis on the right-hand side represents the Hamiltonian operator ($\hat{H}$), which is the total energy operator of the system. So, the TDSE can be written more compactly as:

$i\hbar \frac{\partial}{\partial t} \Psi(x, t) = \hat{H} \Psi(x, t)$

Time-Independent Schrödinger Equation (TISE)

The TISE is used for systems where the potential energy $V$ does not depend on time. In such cases, the wave function can be separated into spatial and temporal parts: $\Psi(x, t) = \psi(x) \phi(t)$. The TISE then becomes an eigenvalue equation for the Hamiltonian operator:

$\hat{H} \psi(x) = E \psi(x)$

Or, in its expanded form:

$-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x)$

Where:

• $\psi(x)$ is the spatial part of the wave function
• $E$ is the total energy of the system, which is a constant (eigenvalue)

The solutions to the TISE, $\psi(x)$, are called eigenfunctions, and the corresponding energy values, $E$, are called eigenvalues.

Applications of the Schrödinger Equation

1. Free Particle

A free particle is one that is not subject to any forces, meaning the potential energy $V(x) = 0$. For a free particle, the TISE becomes:

$-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} = E \psi(x)$

This is a second-order linear differential equation. The general solution is of the form:

$\psi(x) = A e^{ikx} + B e^{-ikx}$

Where $k = \frac{\sqrt{2mE}}{\hbar}$.

The term $e^{ikx}$ represents a wave traveling in the positive x-direction, and $e^{-ikx}$ represents a wave traveling in the negative x-direction. The energy $E$ can be any positive value, meaning a free particle can have any energy. The wave function is not localized, reflecting the fact that the particle's position is completely uncertain.

The corresponding time-dependent wave function is:

$\Psi(x, t) = A e^{i(kx - \omega t)} + B e^{-i(kx + \omega t)}$

Where $\omega = E/\hbar$.

2. Particle in a One-Dimensional Infinite Potential Well

Consider a particle confined to a region of space, say between $x=0$ and $x=L$, with infinitely high potential barriers outside this region. The potential is defined as:

$V(x) = \begin{cases} 0 & \text{for } 0 \le x \le L \\ \infty & \text{otherwise} \end{cases}$

Inside the well ($0 < x < L$), $V(x) = 0$, so the TISE is:

$-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} = E \psi(x)$

The general solution is $\psi(x) = A \sin(kx) + B \cos(kx)$, where $k = \frac{\sqrt{2mE}}{\hbar}$.

Now we apply boundary conditions. Since the potential is infinite outside the well, the probability of finding the particle there is zero, meaning $\psi(x) = 0$ for $x \le 0$ and $x \ge L$.

• At $x=0$: $\psi(0) = A \sin(0) + B \cos(0) = B$. Since $\psi(0) = 0$, we must have $B=0$.
• At $x=L$: $\psi(L) = A \sin(kL)$. Since $\psi(L) = 0$ and $A$ cannot be zero (otherwise $\psi(x)$ would be zero everywhere), we must have $\sin(kL) = 0$.

This implies $kL = n\pi$, where $n$ is a positive integer ($n=1, 2, 3, ...$). Thus, $k_n = \frac{n\pi}{L}$.

The allowed values of energy (eigenvalues) are:

$E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{\hbar^2}{2m} \left(\frac{n\pi}{L}\right)^2 = \frac{n^2\pi^2\hbar^2}{2mL^2}$

The corresponding wave functions (eigenfunctions) are:

$\psi_n(x) = A \sin\left(\frac{n\pi x}{L}\right)$

To find $A$, we normalize the wave function, requiring that the total probability of finding the particle in the well is 1:

$\int_0^L |\psi_n(x)|^2 dx = 1$

$\int_0^L A^2 \sin^2\left(\frac{n\pi x}{L}\right) dx = 1$

Solving this integral yields $A^2 \frac{L}{2} = 1$, so $A = \sqrt{\frac{2}{L}}$.

The normalized wave functions are:

$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$ for $0 \le x \le L$, and $\psi_n(x) = 0$ otherwise.

Note that the energy levels are quantized, meaning only discrete energy values are allowed. The lowest energy state ($n=1$) is called the ground state, and it has a non-zero energy ($E_1 = \frac{\pi^2\hbar^2}{2mL^2}$), known as zero-point energy.

3. Barrier Penetration (Tunneling)

Quantum mechanics predicts that a particle can penetrate a potential barrier even if its energy is less than the height of the barrier. This phenomenon is known as tunneling. Consider a potential barrier of height $V_0$ and width $a$, with a particle of energy $E < V_0$ incident on it from the left.

In the region of the barrier ($V_0 > E$), the TISE is:

$-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V_0 \psi(x) = E \psi(x)$

Rearranging, we get:

$\frac{d^2\psi(x)}{dx^2} = \frac{2m(V_0 - E)}{\hbar^2} \psi(x)$

Let $\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$. Then the equation becomes $\frac{d^2\psi(x)}{dx^2} = \kappa^2 \psi(x)$.

The solutions in this region are of the form $\psi(x) = C e^{\kappa x} + D e^{-\kappa x}$.

If the barrier is wide ($a$ is large), the $e^{\kappa x}$ term (growing exponentially) becomes negligible, and the wave function decays exponentially inside the barrier: $\psi(x) \approx D e^{-\kappa x}$.

When the wave function reaches the other side of the barrier, there is a non-zero amplitude, meaning there is a finite probability that the particle will appear on the other side. This probability is related to the transmission coefficient $T$. For a thin barrier, $T \approx e^{-2\kappa a}$.

Tunneling is crucial for many physical phenomena, including radioactive decay (alpha decay), nuclear fusion in stars, and the operation of scanning tunneling microscopes (STMs).

Interpretations and Properties of the Wave Function

4. Probability Interpretation (Born Rule)

The wave function $\Psi(x, t)$ itself is not directly observable. However, its square magnitude, $|\Psi(x, t)|^2$, represents the probability density of finding the particle at position $x$ at time $t$.

$P(x, t) = |\Psi(x, t)|^2 = \Psi^*(x, t) \Psi(x, t)$

Where $\Psi^*(x, t)$ is the complex conjugate of $\Psi(x, t)$.

The probability of finding the particle in a small interval $dx$ around $x$ at time $t$ is $P(x, t) dx$.

For the probability to be physically meaningful, the wave function must be normalized, meaning the total probability of finding the particle *somewhere* must be 1:

$\int_{-\infty}^{\infty} |\Psi(x, t)|^2 dx = 1$

5. Expectation Values

In quantum mechanics, we often talk about the average value or expectation value of an observable quantity (like position, momentum, or energy). The expectation value of an observable represented by an operator $\hat{A}$ is given by:

$\langle A \rangle = \int_{-\infty}^{\infty} \Psi^*(x, t) \hat{A} \Psi(x, t) dx$

For example:

• Expectation value of position: $\langle x \rangle = \int_{-\infty}^{\infty} x |\Psi(x, t)|^2 dx$
• Expectation value of momentum: $\langle p \rangle = \int_{-\infty}^{\infty} \Psi^*(x, t) (-i\hbar \frac{\partial}{\partial x}) \Psi(x, t) dx$
• Expectation value of energy: $\langle E \rangle = \int_{-\infty}^{\infty} \Psi^*(x, t) \hat{H} \Psi(x, t) dx = E$ (for stationary states)

The expectation value is the average outcome if the measurement were repeated many times on identically prepared systems.

6. Eigenfunctions and Eigenvalues

As mentioned earlier, the solutions to the time-independent Schrödinger equation ($ \hat{H} \psi(x) = E \psi(x) $) are called eigenfunctions ($\psi(x)$) and eigenvalues ($E$).

Eigenfunctions represent states of the system with a definite value of the observable corresponding to the operator. For the Hamiltonian operator, the eigenfunctions are states with a definite energy $E$.

A key property is that for a given operator, its eigenfunctions form a complete set. This means any arbitrary wave function can be expressed as a linear combination (superposition) of these eigenfunctions.

For example, if $\psi_n(x)$ are the eigenfunctions of the Hamiltonian with eigenvalues $E_n$, any state $\Psi(x, t)$ can be written as:

$\Psi(x, t) = \sum_n c_n \psi_n(x) e^{-iE_n t/\hbar}$

Where $c_n$ are complex coefficients determined by the initial state. The probability of measuring the energy $E_n$ for this state is $|c_n|^2$.

7. Stationary States

Stationary states are states whose probability density $|\Psi(x, t)|^2$ does not change with time. These are the states described by the solutions to the time-independent Schrödinger equation.

If a system is in an eigenstate of the Hamiltonian $\psi_n(x)$ with energy $E_n$, its time-dependent wave function is:

$\Psi_n(x, t) = \psi_n(x) e^{-iE_n t/\hbar}$

The probability density for this state is:

$|\Psi_n(x, t)|^2 = |\psi_n(x) e^{-iE_n t/\hbar}|^2 = \psi_n^*(x) e^{iE_n t/\hbar} \psi_n(x) e^{-iE_n t/\hbar} = |\psi_n(x)|^2$

This is independent of time. Systems in stationary states have a definite energy $E_n$.

8. Wave Packets

A single plane wave (like $e^{ikx}$) represents a particle with definite momentum but completely uncertain position. To describe a particle that is somewhat localized in space, we superimpose (add together) several plane waves with different wave numbers $k$ (and thus different momenta and energies).

This superposition of waves creates a "wave packet." In regions where the waves interfere constructively, the amplitude is large, and the probability of finding the particle is high. In regions where they interfere destructively, the amplitude is small, and the probability is low.

The group velocity ($v_g$) of the wave packet represents the speed at which the packet's overall envelope moves, and it corresponds to the classical velocity of the particle. For a free particle, $E = \frac{\hbar^2 k^2}{2m}$, and $p = \hbar k$. The group velocity is $v_g = \frac{d\omega}{dk} = \frac{dE/\hbar}{dp/\hbar} = \frac{dE}{dp}$.

$v_g = \frac{d}{dk} \left(\frac{\hbar k^2}{2m}\right) = \frac{\hbar (2k)}{2m} = \frac{\hbar k}{m} = \frac{p}{m}$. This is precisely the classical velocity.

Wave packets spread out over time because the different component waves travel at slightly different speeds.

9. Uncertainty Principle

The Heisenberg Uncertainty Principle is a fundamental consequence of the wave nature of particles. It states that certain pairs of physical properties, like position and momentum, cannot both be known with arbitrary precision simultaneously.

Mathematically, for position ($x$) and momentum ($p_x$), the principle is stated as:

$\Delta x \Delta p_x \ge \frac{\hbar}{2}$

Where:

• $\Delta x$ is the uncertainty in position
• $\Delta p_x$ is the uncertainty in momentum along the x-axis
• $\hbar$ is the reduced Planck constant

This means if you know the position of a particle very precisely ($\Delta x$ is small), then its momentum must be very uncertain ($\Delta p_x$ is large), and vice versa.

The uncertainty principle arises directly from the properties of wave packets. A perfectly localized wave packet ($\Delta x = 0$) would require an infinite range of wavelengths (and thus momenta) to construct, leading to infinite uncertainty in momentum. Conversely, a wave with a single wavelength ($\Delta p_x = 0$) extends infinitely in space ($\Delta x = \infty$).

There are similar uncertainty relations for other pairs of non-commuting observables, such as energy and time:

$\Delta E \Delta t \ge \frac{\hbar}{2}$

This relation implies that the energy of a system can only be known precisely if the system exists for a long time. Short-lived states have inherently uncertain energies.