The Schrödinger Equation in Dirac Notation

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

The physical state of a system is represented by a set of probability amplitudes (wave functions), which form a linear vector space. This linear vector space is a particular type of space called a Hilbert Space. Another way to think about the Hilbert space is as an infinite-dimensional space of square normalizable functions. This is analogous to a three-dimensional space, where the basis is $\left({\hat {i}},{\hat {j}},{\hat {k}}\right)$ in a generalized coordinate system. In the Hilbert space, the basis is formed by an infinite set of complex functions. The basis for a Hilbert space is written like $\left(|\psi _{0}\rangle ,|\psi _{1}\rangle ,|\psi _{2}\rangle ,...,|\psi _{j}\rangle ,...\right)$, where each $|\psi _{i}\rangle$ is a complex vector function.

We denote a state vector $\psi \$ in Hilbert space with Dirac notation as a “ket” $|\psi \rangle$, and its complex conjugate (or dual vector) $\psi ^{\ast }$ is denoted by a “bra” $\langle \psi |$.

Therefore, in the space of wavefunctions that belong to the Hilbert space, any wave function can be written as a linear combination of the basis functions:

$|\phi \rangle =\sum _{n}c_{n}|\psi _{n}\rangle ,$

where $c_{n}$ is a complex number.

By projecting the state vector $|\psi \rangle$ onto different bases, we can obtain the wave functions of the system in different bases. For example, if we project $|\psi \rangle$ onto the position basis $\langle {\textbf {r}}|,$ we would get $\langle {\textbf {r}}|\psi \rangle \equiv \psi ({\textbf {r}}),$ while projecting onto the momentum basis $\langle {\textbf {p}}|$ gives us $\langle {\textbf {p}}|\psi \rangle \equiv \phi ({\textbf {p}}).$ We interpret $|\psi ({\textbf {r}})|^{2}$ as the probability density of finding the system at position ${\textbf {r}},$ and $|\phi ({\textbf {p}})|^{2}$ as the probability density of finding the system with momentum ${\textbf {p}}$.

In Dirac notation, the scalar product of two state vectors $|\phi \rangle$ and $|\psi \rangle$ is denoted by a “bracket” $\langle \phi |\psi \rangle$. In the position-space representation, the scalar product is given by

$\langle \phi |\psi \rangle =\int \phi ^{\ast }(\mathbf {r} ,t)\psi (\mathbf {r} ,t)\,d^{3}\mathbf {r} ,$

and thus the normalization condition may now be written as

$\langle \psi _{m}|\psi _{n}\rangle =\delta _{mn}.$

This additionally shows that any wave function is determined to within a phase factor, $e^{i\gamma }$, where $\gamma$ is some real number.

The vectors in this space also obey some useful rules following from the fact that the Hilbert space is linear and complete:

$\langle \phi |c\psi \rangle =c\langle \phi |\psi \rangle$

$\langle c\phi |\psi \rangle =c^{*}\langle \phi |\psi \rangle$ where $c$ is a complex number.

$\langle \phi |\psi _{1}+\psi _{2}\rangle =\langle \phi |\psi _{1}\rangle +\langle \phi |\psi _{2}\rangle$

In Dirac notation, the Schrödinger equation is written as

$i\hbar {\frac {d}{dt}}|\psi (t)\rangle ={\mathcal {H}}|\psi (t)\rangle$

By projecting the equation in position space, we can obtain the previous form of the Schrödinger equation,

$i\hbar {\frac {\partial \psi ({\textbf {r}},t)}{\partial t}}=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\textbf {r}})\right]\psi ({\textbf {r}},t).$

On the other hand, we can also project it into momentum space and obtain

$i\hbar {\frac {\partial \phi ({\textbf {p}},t)}{\partial t}}=\left[{\frac {{\textbf {p}}^{2}}{2m}}+V\left(i\hbar {\frac {\partial }{\partial {\textbf {p}}}}\right)\right]\phi ({\textbf {p}},t),$

where $\phi ({\textbf {p}},t)$ and $\psi ({\textbf {r}},t)$ are related through Fourier transform as described in the next section.

For time-independent Hamiltonians, the wave function may be separated into a position-dependent part and a time-dependent part,

$|\psi _{n}(t)\rangle =e^{-iE_{n}t/\hbar }|\psi _{n}\rangle$.

as described previously, thus yielding the equation for stationary states in Dirac notation:

$E_{n}|\psi _{n}\rangle ={\mathcal {H}}|\psi _{n}\rangle .$

The eigenfunctions (now also referred to as eigenvectors) are replaced by eigenkets. Use of this notation makes solution of the Schrödinger equation much simpler for some problems, where the Hamiltonian can be re-written in the form of matrix operators having some algebra (defined set of operations on the basis vectors) over the Hilbert space of the eigenvectors of that Hamiltonian. (See the section on operators.)

We now ask how an arbitrary state $|\phi \rangle$ evolves in time? The initial state $|\phi \rangle$ can be expressed as the linear superposition of the energy eignstates:

$|\phi \rangle =\sum _{n}c_{n}|\psi _{n}\rangle$

We can then solve the time-dependent Schrödinger equation, we obtain, for a time-independent Hamiltonian,

$|\phi (t)\rangle =e^{-i{\mathcal {H}}t/\hbar }|\phi \rangle =e^{-i{\mathcal {H}}t/\hbar }\sum _{n}c_{n}|\psi _{n}\rangle =\sum _{n}c_{n}e^{-iE_{n}t/\hbar }|\psi _{n}\rangle .$