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Schrödinger’s wave equation∗
Cosmin†
2013-03-21 18:43:53
wave equation
The Schrödinger wave equation is considered to be the most basic equation
of non-relativistic quantum mechanics. In three spatial dimensions (that is, in
R3 ) and for a single particle of mass m, moving in a field of potential energy V ,
the equation is
∂
~2
Ψ(r, t) = −
· 4 Ψ(r, t) + V (r, t) Ψ(r, t),
∂t
2m
where r := (x, y, z) is the position vector, ~ = h(2π)−1 , h is Planck’s constant,
4 denotes the Laplacian and V (r, t) is the value of the potential energy at point
r and time t. This equation is a second order homogeneous partial differential
equation which is used to determine Ψ, the so-called time-dependent wave function, a complex function which describes the state of a physical system at a
certain point r and a time t (Ψ is thus a function of 4 variables: x, y, z and
t). The right hand side of the equation represents in fact the Hamiltonian operator (or energy operator) HΨ(r, t), which is represented here as the sum of
the kinetic energy and potential energy operators. Informally, a wave function
encodes all the information that can be known about a certain quantum mechanical system (such as a particle). The function’s main interpretation is that
of a position probability density for the particle1 (or system) it describes, that
is, if P (r, t) is the probability that the particle is at position r at time t, then
an important postulate of M. Born states that P (r, t) = |Ψ(r, t)|2 .
An example of a (relatively simple) solution of the equation is given by the
wave function of an arbitrary (non-relativistic) free2 particle (described by a
wave packet which is obtained by superposition of fixed momentum solutions of
the equation). This wave function is given by:
Z
2
−1
Ψ(r, t) =
A(k)ei(k·r−~k (2m) t) dk,
i~
K
∗ hSchrodingersWaveEquationi
created: h2013-03-21i by: hCosmini version: h36756i
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1 This is in fact a little imprecise since the wave function is, in a way, a statistical tool: it
describes a large number of identical and identically prepared systems. We speak of the wave
function of one particle for convenience.
2 By free particle, we imply that the field of potential energy V is everywhere 0.
1
where k is the wave vector and K is the set of all values taken by k. For a free
particle, the equation becomes
i~
~2
∂
Ψ(r, t) = −
· 4 Ψ(r, t)
∂t
2m
and it is easy to check that the aforementioned wave function is a solution.
An important special case is that when the energy E of the system does
not depend on time, i.e. HΨ = EΨ, which gives rise to the time-independent
Schrödinger equation:
~2
· 4 Ψ(r) + V (r) Ψ(r).
2m
There are a number of generalizations of the Schrödinger equation, mostly in
order to take into account special relativity, such as the Dirac equation (which
describes a spin- 12 particle with mass) or the Klein-Gordon equation (describing
spin-0 particles).
EΨ(r) = −
2