The Schrödinger equation is the foundation of quantum mechanics, describing how particles behave at the atomic level. It comes in two flavors: time-dependent and time-independent. These equations help us understand the wave-like nature of matter and predict the behavior of quantum systems.

The time-dependent version shows how a particle's state changes over time, while the time-independent version deals with stationary states. Together, they give us a complete picture of quantum systems, from simple particles in boxes to complex molecules and materials.

Derivation of Schrödinger's Equation

Time-Dependent Schrödinger Equation

• The time-dependent Schrödinger equation is derived from the postulate that the state of a quantum system is completely described by a wave function $\Psi(r, t)$ that satisfies the equation $i\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi$, where $\hat{H}$ is the Hamiltonian operator
• The Hamiltonian operator $\hat{H}$ is the sum of the kinetic energy operator $\hat{T}$ and the potential energy operator $\hat{V}$, i.e., $\hat{H} = \hat{T} + \hat{V}$
• The kinetic energy operator is given by $\hat{T} = -\frac{\hbar^2}{2m}\nabla^2$, where $\nabla^2$ is the Laplacian operator and $m$ is the mass of the particle
• The potential energy operator $\hat{V}$ depends on the specific system and represents the external forces acting on the particle (harmonic oscillator potential, Coulomb potential)

Time-Independent Schrödinger Equation

• The time-independent Schrödinger equation is derived by assuming that the wave function can be separated into a time-dependent part and a space-dependent part, leading to the equation $\hat{H}\Psi(r) = E\Psi(r)$, where $E$ is the energy eigenvalue
• The time-independent Schrödinger equation is an eigenvalue problem, where the wave function $\Psi(r)$ is an eigenfunction of the Hamiltonian operator with a corresponding energy eigenvalue $E$
• Solutions to the time-independent Schrödinger equation represent stationary states, which are quantum states with well-defined energy (ground state, excited states)

Solving Schrödinger's Equation for Simple Systems

Particle in a Box

• The particle in a box model assumes a particle confined within a one-dimensional box of length $L$ with infinite potential walls
• The time-independent Schrödinger equation for a particle in a box is solved by applying boundary conditions, i.e., the wave function must be zero at the walls ($\Psi(0) = \Psi(L) = 0$)
• The energy eigenvalues for a particle in a box are given by $E_n = \frac{n^2h^2}{8mL^2}$, where $n$ is a positive integer (quantum number)
• The corresponding wave functions for a particle in a box are given by $\Psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L})$
• The particle in a box model demonstrates the quantization of energy and the formation of standing waves (nodes, antinodes)

Harmonic Oscillator

• The harmonic oscillator model describes a particle subject to a quadratic potential $V(x) = \frac{1}{2}kx^2$, where $k$ is the spring constant
• The time-independent Schrödinger equation for a harmonic oscillator is solved using mathematical techniques such as the power series method or the creation and annihilation operator approach
• The energy eigenvalues for a harmonic oscillator are given by $E_n = (n + \frac{1}{2})\hbar\omega$, where $\omega = \sqrt{\frac{k}{m}}$ is the angular frequency and $n$ is a non-negative integer (quantum number)
• The corresponding wave functions for a harmonic oscillator are given by the Hermite polynomials multiplied by a Gaussian function
• The harmonic oscillator model is used to describe the vibrational motion of molecules and the behavior of phonons in solids

Interpretation of the Wavefunction

Probability Density

• The wave function $\Psi(r, t)$ is a complex-valued function that contains all the information about the quantum state of a system
• The physical meaning of the wave function is that its squared modulus, $|\Psi(r, t)|^2$, represents the probability density of finding the particle at a specific position $r$ at time $t$
• The probability of finding the particle in a small volume element $dr$ is given by $P(r, t)dr = |\Psi(r, t)|^2dr$
• The wave function is normalized, meaning that the integral of $|\Psi(r, t)|^2$ over all space is equal to 1, ensuring that the total probability of finding the particle somewhere in space is 100%

Phase and Interference

• The phase of the wave function, $\arg(\Psi)$, does not have a direct physical interpretation but plays a crucial role in the interference and superposition of quantum states
• When two or more wave functions overlap, they interfere constructively or destructively depending on their relative phases (double-slit experiment, quantum eraser)
• The interference of wave functions leads to the formation of interference patterns, which demonstrate the wave-like behavior of quantum particles

Properties of Stationary States

Energy Eigenvalues and Eigenfunctions

• Stationary states are solutions to the time-independent Schrödinger equation and represent quantum states with well-defined energy
• The energy eigenvalues are the allowed energy values that a quantum system can possess in a particular stationary state
• The corresponding wave functions, called eigenfunctions, describe the spatial distribution of the particle in each stationary state (ground state, excited states)

Orthogonality and Completeness

• Stationary states are orthogonal, meaning that the integral of the product of two different stationary states over all space is zero
• Orthogonality ensures that stationary states are linearly independent and can be used as a basis for the Hilbert space of the quantum system
• Stationary states form a complete set, allowing any arbitrary state to be expressed as a linear combination of stationary states
• Completeness is essential for the mathematical description of quantum systems and the expansion of wave functions in terms of stationary states

Expectation Values

• The expectation value of an observable in a stationary state is time-independent, as the probability density does not change with time
• Expectation values provide information about the average value of a physical quantity in a given quantum state (position, momentum, energy)
• The expectation value of an observable $\hat{A}$ in a state $\Psi$ is calculated using the integral $\langle \hat{A} \rangle = \int \Psi^* \hat{A} \Psi dr$, where $\Psi^*$ is the complex conjugate of the wave function

Time Evolution of Quantum Systems

Time-Dependent Solutions

• The time-dependent Schrödinger equation describes how the wave function $\Psi(r, t)$ evolves over time under the influence of the Hamiltonian operator
• For a time-independent Hamiltonian, the solution to the time-dependent Schrödinger equation can be expressed as a linear combination of stationary states, i.e., $\Psi(r, t) = \sum_n c_n \Psi_n(r) \exp(-iE_nt/\hbar)$, where $c_n$ are complex coefficients
• The coefficients $c_n$ represent the probability amplitudes of finding the system in each stationary state and are determined by the initial conditions
• The time evolution of the wave function leads to the phenomenon of quantum beats, where the probability density oscillates with frequencies related to the energy differences between the stationary states

Interaction with External Fields

• The time-dependent Schrödinger equation is essential for understanding time-dependent processes such as the interaction of a quantum system with an external field (electric field, magnetic field)
• When a quantum system interacts with an external field, the Hamiltonian operator becomes time-dependent, and the wave function evolves according to the modified Schrödinger equation
• The interaction with external fields can induce transitions between different stationary states, leading to phenomena such as absorption, emission, and Rabi oscillations (atoms in laser fields, nuclear magnetic resonance)

Quantum Dynamics

• The time-dependent Schrödinger equation describes the dynamics of a quantum system undergoing a change in the potential energy (quantum tunneling, scattering)
• Quantum dynamics involves the study of time-dependent processes, such as the motion of wave packets, the spreading of probability distributions, and the coherent control of quantum states
• Numerical methods, such as the split-operator method and the finite-difference time-domain method, are used to solve the time-dependent Schrödinger equation for complex systems (molecules, nanostructures)