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**Consider a particle in a time-independent scalar potential.**

The Schroedinger equation for such a particle is iħ∂Ψ(**r**,t)/∂t
= -(ħ^{2}/(2m))∇^{2}Ψ(**r**,t) + U(**r**)Ψ(**r**,t).

The wave function is a function of position **r** and time t.**Is separation of variables possible?**

Assume solutions of the form** **Ψ(**r**,t) = Φ(**r**)ζ(t)
exist.

Inserting Ψ(**r**,t) = Φ(**r**)ζ(t) into the Schroedinger
equation we obtain

(iħ/ζ(t)) ∂ζ(t)/∂t =
(1/Φ(**r**))[-(ħ^{2}/(2m))∇^{2}Φ(**r**)
] + U(**r**)Φ(**r**).

The left hand side of the equation is a function of t, independent of **r**,
and the right hand side is a function of **r**, independent of t.
Both
sides must therefore be equal to a constant, which we call E. The left hand side
then yields

iħ∂ζ(t))/∂t = Eζ(t). ζ(t) = A
exp(-iEt/ħ) = A exp(-iωt).

Here ω is the frequency of the wave function solution, and
the de Broglie relation identifies E as the energy of the particle.

The right hand
side yields H Φ(**r**) = E Φ(**r**),
with H = -(ħ^{2}/(2m))∇^{2} + U(**r**) being a differential operator, or

∇^{2}Φ(**r**) + (2m/ħ^{2})(E -
U(**r**))Φ(**r**) = 0.

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**Summary**

If U = U(**r**), then
solutions of the type Φ(**r**)ζ(t) exist. They are
called **stationary solutions** or **stationary states**.
They correspond to states in which the
particle has a well defined energy E =
ħω.

Φ(**r**) is a solution of HΦ(**r**) = EΦ(**r**),
the **time-independent
Schroedinger equation**. It is an energy eigenfunction, since the time-independent Schroedinger equation is an **eigenvalue equation** of the linear operator H.
The
allowed energies are called the eigenvalues of H.

If solutions Φ_{n}(**r**) can
be found, for different values of E_{n}, then the linear combination

Ψ(**r**,t) = ∑_{n }c_{n}Φ_{n}(**r**)exp(-iE_{n}t/ħ)

is also a solution of the Schroedinger
equation, because the Schroedinger equation is a linear equation.

However, such a linear
combination does not represent a particle with a well defined energy.