Schrodinger equation

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The Schrodinger equation is a linear differential equation used in various fields of physics to describe the time evolution of quantum states. It is a fundamental aspect of quantum mechanics. The equation is named for its discoverer, Erwin Schrodinger.

Contents

Mathematical forms

General time-dependent form

The Schrodinger equation may generally be written

i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle


where i is the imaginary unit,
\hbar is Planck's constant divided by ,
|\Psi\rangle is the quantum mechanical state or wavefunction (expressed here in Dirac notation), and
\hat H is the Hamiltonian operator.

The left side of the equation describes how the wavefunction changes with time; the right side is related to its energy. For the simplest case of a particle of mass m moving in a one-dimensional potential V(x), the Schrodinger equation can be written


-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}

Derivation

The quickest and easiest way to derive Schrodinger's equation is to understand the Hamiltonian operator in quantum mechanics. In classical mechanics, the total energy of a system is given by


E = \frac{p^2}{2m} + V(x)

where p is the momentum of the particle and V(x) is its potential energy. Applying the quantum mechanical operator for momentum:


p = \frac{\hbar}{i}\frac{\partial}{\partial x}

and subbing into the classical mechanical form for energy, we get the same Hamiltonian operator in quantum mechanics:


\hat H = \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)

from which Schrodinger's equation and the eigenvalue problem  \hat H\Psi = E\Psi can be easily seen.

Eigenvalue problems

In many instances, steady-state solutions to the equation are of great interest. Physically, these solutions correspond to situations in which the wavefunction has a well-defined energy. The energy is then said to be an eigenvalue for the equation, and the wavefunction corresponding to that energy is called an eigenfunction or eigenstate. In such cases, the Schrodinger equation is time-independent and is often written

E\psi=\hat H\psi

Here, E is energy, H is once again the Hamiltonian operator, and ψ is the energy eigenstate for E.

One example of this type of eigenvalue problem is an electrons bound inside an atom.

Examples for the time-independent equation

Free particle in one dimension

In this case, V(x) = 0 and so we see that the solution to the Schrodinger equation must be

ψ = Aeikx

with energy given by

E=\frac{\hbar^2 k^2}{2m}

Physically, this corresponds to a wave travelling with a momentum given by \hbar k, where k can in principle take any value.

Particle in a box

Consider a one-dimensional box of width a, where the potential energy is 0 inside the box and infinite outside of it. This means that ψ must be zero outside the box. One can verify (by substituting into the Schrodinger equation) that

ψ = sin(kx)

is a solution if k = nπ where n is any integer. Thus, rather than the continuum of solutions for the free particle, for the particle in a box there is a set of discrete solutions with energies given by

E_n=\frac{\hbar^2 k^2}{2m}=\frac{\hbar^2n^2\pi^2}{2m}

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