2.3 Schrödinger equation and wave functions

Quantum mechanics revolves around the Schrödinger equation and wave functions. These mathematical tools describe how particles behave at the quantum level, predicting their positions and energies over time.

The Schrödinger equation comes in time-dependent and time-independent forms. Wave functions represent a particle's quantum state, with their square giving the probability of finding the particle at a specific location.

Schrödinger Equation and Wave Functions

Schrödinger equations: Time dependence

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• Time-dependent Schrödinger equation describes the evolution of a quantum system over time represented by the equation $i\hbar\frac{\partial}{\partial t}\Psi(x,t) = \hat{H}\Psi(x,t)$
  • $i$ represents the imaginary unit
  • $\hbar$ represents the reduced Planck's constant (fundamental constant in quantum mechanics)
  • $\Psi(x,t)$ represents the wave function dependent on position $x$ and time $t$
  • $\hat{H}$ represents the Hamiltonian operator (total energy operator)
• Time-independent Schrödinger equation describes the stationary states of a quantum system
  • Obtained by separating the time and space variables in the time-dependent equation
  • Represented by the equation $\hat{H}\psi(x) = E\psi(x)$
    • $\psi(x)$ represents the time-independent wave function dependent only on position $x$
    • $E$ represents the energy eigenvalue (constant value associated with a specific stationary state)

Wave function and probability density

• Wave function $\Psi(x,t)$ is a complex-valued function that contains all the information about a quantum system
  • Probability amplitude: the wave function itself is not a probability, but its square modulus is related to probability (Born's interpretation)
  • Example: the wave function of an electron in an atom describes its quantum state
• Probability density is given by $|\Psi(x,t)|^2$ and represents the probability of finding a particle at a specific position $x$ at time $t$
  • Normalized wave function ensures that the total probability of finding the particle somewhere in space is 1, represented by $\int_{-\infty}^{\infty}|\Psi(x,t)|^2dx = 1$
  • Example: the probability density of an electron in a hydrogen atom is highest near the nucleus

Solving the Schrödinger Equation and Wave Function Properties

Solutions for simple quantum systems

• Particle in a one-dimensional infinite potential well (particle in a box) is a simple quantum system with potential energy $V(x) = 0$ for $0 < x < L$, and $V(x) = \infty$ elsewhere
  • Time-independent Schrödinger equation for this system: $-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi(x) = E\psi(x)$
  • Solutions (energy eigenstates) are given by $\psi_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L})$, where $n = 1, 2, 3, ...$
  • Energy eigenvalues are quantized and given by $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$
  • Example: an electron confined in a nanoscale structure (quantum dot) can be modeled as a particle in a box

Properties of wave functions

• Normalization ensures that the total probability of finding the particle somewhere in space is 1
  • For a normalized wave function, $\int_{-\infty}^{\infty}|\Psi(x,t)|^2dx = 1$
  • Example: the wave function of a particle must be normalized to ensure conservation of probability
• Orthogonality: two wave functions $\psi_i(x)$ and $\psi_j(x)$ are orthogonal if their inner product is zero
  • Inner product is defined as $\langle\psi_i|\psi_j\rangle = \int_{-\infty}^{\infty}\psi_i^*(x)\psi_j(x)dx = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta (1 if $i = j$ and 0 otherwise)
  • Orthonormal set is a set of wave functions that are both orthogonal and normalized
  • Example: the energy eigenstates of a quantum harmonic oscillator form an orthonormal set