Download CR2

LECTURE-2
Schrödinger equation
In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the
quantum state of a physical system changes with time. It was formulated in late 1925, and published in 1926,
by the Austrian physicist Erwin Schrödinger.
In classical mechanics, the equation of motion is Newton's second law, (F
= ma), used to mathematically
predict what the system will do at any time after the initial conditions of the system. In quantum mechanics,
the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and
subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a
linear partial differential equation, describing the time-evolution of the system's wave function (also called a
"state function").
The concept of a wavefunction is a fundamental postulate of quantum mechanics. Schrödinger's equation is
also often presented as a separate postulate, but some authors assert it can be derived from symmetry
principles. Generally, "derivations" of the SE demonstrate its mathematical plausibility for describing wave–
particle duality.
Time-dependent equation
The form of the Schrödinger equation depends on the physical situation (see below for special cases). The
most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving
with time:
Time-dependent Schrödinger equation (general)
where i is the imaginary unit, ħ is the Planck constant divided by 2π, the symbol ∂/∂t indicates a partial
derivative with respect to time t, Ψ (the Greek letter Psi) is the wave function of the quantum system, and Ĥ
is the Hamiltonian operator (which characterizes the total energy of any given wave function and takes
different forms depending on the situation).
A wave function that satisfies the non-relativistic Schrödinger equation with V = 0. In other words, this
corresponds to a particle traveling freely through empty space. The real part of the wave function is
plotted here.
The most famous example is the non-relativistic Schrödinger equation for a single particle moving in an
electric field (but not a magnetic field; see the Pauli equation):
Time-dependent Schrödinger equation
(single non-relativistic particle)
where μ is the particle's "reduced mass", V is its potential energy, ∇2 is the Laplacian, and Ψ is the wave
function (more precisely, in this context, it is called the "position-space wave function"). In plain language, it
means "total energy equals kinetic energy plus potential energy", but the terms take unfamiliar forms for
reasons explained below.
Given the particular differential operators involved, this is a linear partial differential equation. It is also a
diffusion equation, but unlike the heat equation, this one is also a wave equation given the imaginary unit
present in the transient term.
The time-independent Schrödinger equation is the equation describing stationary states. (It is only used when
the Hamiltonian itself is not dependent on time. In general, the wave function still has a time dependency.)
Time-independent Schrödinger equation (general)
In words, the equation states:
When the Hamiltonian operator acts on a certain wave function Ψ, and the result is proportional
to the same wave function Ψ, then Ψ is a stationary state, and the proportionality constant, E,
is the energy of the state Ψ.
The time-independent Schrödinger equation is discussed further below. In linear algebra terminology, this
equation is an eigenvalue equation.
As before, the most famous manifestation is the non-relativistic Schrödinger equation for a single particle
moving in an electric field (but not a magnetic field):
Time-independent Schrödinger equation (single non-relativistic particle)