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 Schrödinger equation, as introduced in the previous chapter, is a special case of a more general equation that is satisfied by the abstract state vector $|\Psi (t)\rangle$ describing the system. More specifically, it is an equation describing the components of the state vector in position space. We will now introduce this more general equation, written in terms of the state vector itself, and show how one can recover the wave equation from the previous chapter.

In Dirac notation, the Schrödinger equation is written as

$i\hbar {\frac {d}{dt}}|\Psi (t)\rangle ={\hat {H}}(t)|\Psi (t)\rangle .$

We see that the Hamiltonian of the system determines how a given initial state will evolve in time.

To show how to recover the equation for the wave function, let us consider the Hamiltonian for a particle moving in one dimension,

${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V({\hat {x}},t).$

We now write our state vector in position space. Since the position space is continuous, rather than discrete, the state vector as a linear superposition of position eigenstates must now be written as an integral:

$|\Psi (t)\rangle =\int dx\,\Psi (x,t)|x\rangle ,$

where $\langle x'|x\rangle =\delta (x'-x)$ and $\delta (x)\!$ is the Dirac delta function.

With the aid of the identity,

${\frac {\partial }{\partial x}}\delta (x'-x)={\frac {\delta (x'-x)}{x'-x}},$

one may verify that

$\langle x'|{\hat {p}}|x\rangle =i\hbar {\frac {\delta (x'-x)}{x'-x}}=i\hbar {\frac {\partial }{\partial x}}\delta (x'-x)$

and that

${\hat {p}}|\Psi (t)\rangle =-i\hbar \int dx\,{\frac {\partial \Psi }{\partial x}}|x\rangle .$

If we now substitute the above form of the Hamiltonian into the Schrödinger equation and project the resulting equation into position space, we will arrive at the wave equation stated in the previous chapter,

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

The above procedure can be generalized to multiple dimensions, again recovering the multi-dimensional wave equation given in the previous chapter:

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

We could also have chosen to work in momentum space; a similar procedure yields

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

Here, $\Phi ({\textbf {p}},t)$ and $\Psi ({\textbf {r}},t)$ are related through a Fourier transform as described in a previous section.

For a time-independent Hamiltonian, we may solve the Schrödinger equation by first assuming that the state vector has the form,

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

as described previously. Substituting this form into the Schrödinger equation yields the equation for stationary states in Dirac notation:

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

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 rewritten 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.

We now ask how an arbitrary state $|\phi \rangle$ evolves in time. The initial state $|\phi \rangle$ can be expressed as a 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,

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