Electromagnetic radiation has the nature of a stream of particles as well as waves. The particles are called photons. As waves the radiation has a frequency ν, expressed in cycles per second. The energy of a photon is equal to hν, where h is Planck's constant. There is another frequency, called the angular frequency, which in radians per second and is the regular freqency times 2π. Let ω denote the angular frequency. Since ω=2πν

E = hν = (h/2π)2πν = hω

Planck's constant divided by 2π is called h-bar.

A wave has a wave length ψ. Wave length and frequency are tied together by their product being equal to the speed of propagation. For electromagnetic radiation the speed of propagation is about 3×10⁸ meter per second. This is called the speed of light in a vacuum and is usually denoted as c. Thus

ψν = c

In terms of the angular frequency ω this above relationship is

(ψ/2π)2πν = (1/κ)ω = c

where κ=2π/ψ=ω/c is called the wave number

The Wave Characteristics of Particles

Louis de Broglie conjectured that just as radiation had particle-like characteristics then matter might have wave-like characteristics. De Broglie further suggested that the wave length λ of a particle is given by

λ = h/p
and consequently
p = hκ

where p is the linear momentum of the particle. This proved to be true.

The Probability Density Function for a Particle

The probability density function for a particle is the probability per unit volume of space and time of finding the particle near any particular location at any particular time. Let (x, y, z) be the space coordinates and t the time coordinate. Then the probability density function φ(x, y, z, t) is such that the probability P of finding the particle in a volume V and in an interval of time T is given by:

P = ∫_V∫_Tφ(x,y,z,t)dVdt

The function φ(x,y,z,t) must be everywhere nonnegative and such that its integration over all space and all time is equal to 1.0. The dimensions of φ are [L^-3T^-1].

The Wave Function for a Particle

In physics it has proven fruitful to work not with the probability density function itself but with a complex-valued function ψ(x,y,z,t) such that

φ(x,y,z,t) = ψ(x,y,z,t)ψ*(x,y,z,t)

where ψ* is the complex conjugate of ψ. This definition guarantees that φ is nonnegative. Note however that the dimensions of ψ are [L^-3/2T^-1/2].

Motion of a Particle in a Region
of Constant Potential Energy

Consider a particle with a momentum vector p moving in a region of constant potential energy. It will move along a straight line in the direction of its momentum vector. Then the propagation vector q is given by

q = p/h

Such motion is described as a plane wave with the following wave function

ψ(r, t) = A*exp(i(k·r − ωt))

where i is the imaginary unit and A is a complex constant. (Note that A could be a real number since the real numbers are special cases of complex numbers.)

In a one dimensional space a particle traveling along the x-axis, at any particular time, the probability density φ=ψψ*=|ψ|² has the form

Since κ=ph and ω=E/h the wave function for a plane wave can be expressed as

ψ(r, t) = A*exp(i(p·r/h − Et/h))

Derivation of the Schroedinger Equation

Now the task is to find the partial differential equation that has the above wave function as a solution.

In Cartesian coordinates the quantity p·r has the form

p·r = p_xx + p_yy + p_zz

The differentiation of ψ once with respect to x gives

∂ψ/∂x = (ip_x/h)ψ

and a second time

∂²ψ/∂x² = (−p_x²/h²)ψ

Likewise the second derivatives of ψ with respect to y and z produces analogous expressions. When these second derivatives are added together the result is

∂²ψ/∂x² + ∂²ψ/∂y² + ∂²ψ/∂z² = − [(p_x²p_y²+p_z²)/h²]ψ
which can be expressed more succinctly as
∇²ψ = −(p²/h²)ψ
and this can be put into the form
p²ψ = −h²∇²ψ

The expression ∇² is called the Laplacian operator. Technically it is the divergence of the gradient of its scalar argument.

The differentiation of the wave function with respect to time gives

∂ψ/∂t = − (iE/h)ψ
which may be expressed as
Eψ = (−1/i)h∂ψ/∂t = ih∂ψ/∂t

The energy E of a particle is the sum of its kinetic energy and its potential energy V; thus

E = p²/2m + V
and hence
Eψ = (1/2m)p²ψ + Vψ

where m is the mass of the particle.

The previously derived expressions for Eψ and p²ψ may be substituted into the above equation to obtain

ih∂ψ/∂t = −h²∇²ψ + Vψ
which may be rearranged
into the form
[−(h²/2m)∇² + V]ψ = ih∂ψ/∂t

This is the Schroedinger Equation.

The expression [−(h²/2m)∇² + V] is usually denoted as an operator H and hence the Schroedinger Equation is expressed in the form

Hψ = ih∂ψ/∂t

This equation from which the wave equation for a particle moving in a field of constant potential energy is also satified for a particle moving in a region of varying potential energy.