Physics:Quantum Schrödinger equation

The Schrödinger equation is the fundamental equation of nonrelativistic quantum mechanics. It governs the time evolution of the wave function of a quantum-mechanical system and provides the mathematical basis for predicting measurable quantities such as probability densities, energies, and stationary states.Template:Rp It was introduced by Erwin Schrödinger in 1925–1926 and became one of the central achievements in the development of modern physics.

The Schrödinger equation is the central equation of wave mechanics and describes the time evolution of a quantum state.

Overview

Conceptually, the Schrödinger equation plays a role in quantum mechanics analogous to that of Newton's second law in classical mechanics. Given an initial quantum state, it determines how that state changes over time.Template:Rp In Schrödinger’s formulation, matter is associated with a wave, following the earlier proposal of Louis de Broglie that particles possess wave-like properties.

The Schrödinger equation is one of several equivalent formulations of quantum mechanics. Other approaches include matrix mechanics, developed by Werner Heisenberg, and the path integral formulation, associated chiefly with Richard Feynman. The Schrödinger approach is often called wave mechanics.

Definition

Preliminaries

In introductory treatments, the Schrödinger equation is often presented in position space for a single nonrelativistic particle in one dimension: $$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }(x,t)=[-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V(x,t)]\mathrm{\Psi }(x,t).$$

Here, $\mathrm{\Psi }(x,t)$ is the wave function, $m$ is the particle mass, and $V(x,t)$ is the potential energy function describing the particle’s environment.Template:Rp The constants $i$ and $\hslash$ denote the imaginary unit and the reduced Planck constant, respectively.Template:Rp

In the more general mathematical formulation of quantum mechanics, a system is described by a state vector $|\psi ⟩$ in a Hilbert space ${\mathscr{H}}$. Physical observables such as position, momentum, and energy are represented by self-adjoint operators acting on this space.

A position-space wave function is obtained from the state vector through $$\mathrm{\Psi }(x,t)=⟨x|\mathrm{\Psi }(t)⟩.$$

Time-dependent Schrödinger equation

The most general form used in ordinary quantum mechanics is the time-dependent Schrödinger equation: $$i\hslash \frac{d}{dt}|\mathrm{\Psi }(t)⟩=\hat{H}|\mathrm{\Psi }(t)⟩$$ where $\hat{H}$ is the Hamiltonian operator of the system.Template:Rp

To apply the equation, one specifies the Hamiltonian of the system, including kinetic and potential energy terms, and solves for the wave function. The square modulus of the wave function gives the probability density: $$\mathrm{Pr}(x,t)=|\mathrm{\Psi }(x,t){|}^2.$$Template:Rp

For a nonrelativistic spinless particle in three dimensions, the position-space form is $$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }(\mathbf{𝐫},t)=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }(\mathbf{𝐫},t)+V(\mathbf{𝐫})\mathrm{\Psi }(\mathbf{𝐫},t).$$

Time-independent Schrödinger equation

When the Hamiltonian does not depend explicitly on time, solutions of definite energy can be found using the time-independent Schrödinger equation: $$\hat{H}|\mathrm{\Psi }⟩=E|\mathrm{\Psi }⟩$$ where $E$ is the energy eigenvalue.Template:Rp

This is an eigenvalue equation: the wave function is an eigenfunction of the Hamiltonian, and the corresponding eigenvalue is the measurable energy. Such solutions describe stationary states, whose probability density does not change with time.

Properties

Linearity

The Schrödinger equation is linear. If $|{\psi }_1⟩$ and $|{\psi }_2⟩$ are solutions, then any linear combination $$|\psi ⟩=a|{\psi }_1⟩+b|{\psi }_2⟩$$ is also a solution, where $a$ and $b$ are complex numbers.Template:Rp This allows for quantum superposition, one of the defining features of quantum theory.

When expressed in an energy eigenbasis, a general time-dependent state may be written as $$|\mathrm{\Psi }(t)⟩=\sum _n{A}_n{e}^{-i{E}_{n}t/\hslash }|{\psi }_{E_n}⟩.$$

Unitarity

Time evolution under the Schrödinger equation is unitary. If the Hamiltonian is time-independent, the formal solution is $$|\mathrm{\Psi }(t)⟩=e^{-i\hat{H}t/\hslash }|\mathrm{\Psi }(0)⟩.$$ The operator $$\hat{U}(t)=e^{-i\hat{H}t/\hslash }$$ is called the time-evolution operator and preserves the inner product in Hilbert space.

As a result, if a state is normalized at one time, it remains normalized at all later times.

Probability current

The Schrödinger equation is consistent with local conservation of probability.Template:Rp In nonrelativistic quantum mechanics, the probability density $\rho (\mathbf{𝐫},t)$ and probability current $\mathbf{𝐣}$ satisfy the continuity equation $$\frac{\partial }{\partial t}\rho (\mathbf{𝐫},t)+\mathrm{\nabla }\cdot \mathbf{𝐣}=0.$$

For a wave function $\psi$, the probability current is $$\mathbf{𝐣}=\frac{\hslash }{m}\mathrm{Im}({\psi }^{*}\mathrm{\nabla }\psi ).$$

Separation of variables

If the potential does not depend explicitly on time, one may seek solutions of the form $$\mathrm{\Psi }(\mathbf{𝐫},t)=\psi (\mathbf{𝐫})\tau (t).$$ This leads to stationary-state solutions $$\mathrm{\Psi }(\mathbf{𝐫},t)=\psi (\mathbf{𝐫})e^{-iEt/\hslash }.$$

The spatial part then satisfies $${\mathrm{\nabla }}^2\psi (\mathbf{𝐫})+\frac{2m}{{\hslash }^2}[E-V(\mathbf{𝐫})]\psi (\mathbf{𝐫})=0.$$

This method is especially useful in systems with symmetry, where Cartesian or spherical coordinates allow further separation.

Examples

Particle in a box

The particle in a box is one of the simplest exactly solvable quantum systems. A particle confined to a region of zero potential with infinite walls outside satisfies $$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}=E\psi .$$

The boundary conditions force the wave function to vanish at the walls, so only certain wavelengths are allowed. This yields quantized energies $$E_n=\frac{{\hslash }^2{\pi }^2{n}^2}{2m{L}^2}=\frac{n^2{h}^2}{8m{L}^2},n=1,2,3,\dots$$ demonstrating that bound states in quantum mechanics have discrete energy levels.Template:Rp

Harmonic oscillator

For the quantum harmonic oscillator, the time-independent Schrödinger equation is $$E\psi =-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}\psi +\frac{1}{2}m{\omega }^2{x}^2\psi .$$

Its energy eigenvalues are $$E_n=(n+\frac{1}{2})\hslash \omega .$$ The lowest state, $n=0$, has nonzero energy, called the zero-point energy.

Hydrogen atom

The Schrödinger equation for the electron in a hydrogen atom is $$E\psi =-\frac{{\hslash }^2}{2\mu }{\mathrm{\nabla }}^2\psi -\frac{q^2}{4\pi {\epsilon }_{0}r}\psi .$$

It can be solved exactly by separation of variables in spherical coordinates. The resulting solutions explain the discrete spectral lines of hydrogen and were among the earliest major successes of wave mechanics.Template:Rp

Approximate solutions

Most quantum systems cannot be solved exactly. Approximate methods such as variational methods, WKB approximation, and perturbation theory are therefore widely used.

Semiclassical limit

The connection between quantum and classical mechanics can be explored through the Ehrenfest theorem, which relates the time evolution of expectation values to classical equations of motion.Template:Rp In the limit where the wave function remains highly localized, quantum expectation values approximately follow classical trajectories.

Writing the wave function as $$\mathrm{\Psi }=\sqrt{\rho (\mathbf{𝐫},t)}{e}^{iS(\mathbf{𝐫},t)/\hslash }$$ and taking the formal limit $\hslash \to 0$ leads to the Hamilton–Jacobi equation, illustrating the correspondence between quantum and classical mechanics.Template:Rp

Density matrix formulation

Wave functions are not always the most convenient way to describe a system. When the preparation is not perfectly known, or when a system is part of a larger one, density matrices are often used instead.Template:Rp The density-matrix analogue of the Schrödinger equation is $$i\hslash \frac{\partial \hat{\rho }}{\partial t}=[\hat{H},\hat{\rho }],$$ where the brackets denote the commutator.

For time-independent Hamiltonians, the solution is $$\hat{\rho }(t)=e^{-i\hat{H}t/\hslash }\hat{\rho }(0)e^{i\hat{H}t/\hslash }.$$

Relativistic generalizations

The ordinary one-particle Schrödinger equation is fundamentally nonrelativistic. In relativistic settings, one instead uses equations such as the Klein–Gordon equation for spin-0 particles and the Dirac equation for spin-Template:Frac particles.

In quantum field theory, the Schrödinger picture can still be formulated, but the Hilbert space is generally a Fock space in which particle number is not fixed.

History

The origins of the Schrödinger equation lie in the development of wave–particle duality. Following Max Planck’s quantization of radiation and Albert Einstein’s light quantum hypothesis, Louis de Broglie proposed that matter particles also have an associated wavelength: $$p=\frac{h}{\lambda }=\hslash k.$$

Building on this idea, Schrödinger sought a wave equation for electrons. His nonrelativistic equation, $$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }(\mathbf{𝐫},t)=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }(\mathbf{𝐫},t)+V(\mathbf{𝐫})\mathrm{\Psi }(\mathbf{𝐫},t),$$ reproduced the observed spectral energies of the hydrogen atom and provided a compelling alternative to earlier quantum models.

Soon afterward, Max Born interpreted the wave function probabilistically, showing that $|\mathrm{\Psi }{|}^2$ gives a probability density rather than a literal charge distribution.Template:Rp

Interpretation

The Schrödinger equation determines how the wave function changes in time, but it does not by itself specify what the wave function is. Different interpretations of quantum mechanics assign different meanings to the wave function and to the measurement process.

In the Copenhagen interpretation, the wave function represents information or probability amplitudes, and measurement involves probabilistic outcomes governed by the Born rule. Other interpretations, including many-worlds and Bohmian mechanics, retain the same Schrödinger equation while differing in how they understand physical reality and measurement.

See also

• Dirac equation
• Klein–Gordon equation
• Particle in a box
• Quantum harmonic oscillator
• Density matrix
• Interpretations of quantum mechanics
• Probability current
• Schrödinger picture

References

Author: Harold Foppele

Source attribution: Physics:Quantum Schrödinger equation