<P> The Schrödinger wave equation, describing states of quantum particles, has solutions that describe a system and determine precisely how the state changes with time . Suppose a wavefunction ψ (x, t) is a solution of the wave equation, giving a description of the particle (position x, for time t). If the wavefunction is square integrable, i.e. </P> <Dl> <Dd> ∫ R n ψ 0 (x, t 0) 2 d x = a 2 <∞ (\ displaystyle \ int _ (\ mathbf (R) ^ (n)) \ psi _ (0) (\ mathbf (x), t_ (0)) ^ (2) \, \ mathrm (d \ mathbf (x)) = a ^ (2) <\ infty) </Dd> </Dl> <Dd> ∫ R n ψ 0 (x, t 0) 2 d x = a 2 <∞ (\ displaystyle \ int _ (\ mathbf (R) ^ (n)) \ psi _ (0) (\ mathbf (x), t_ (0)) ^ (2) \, \ mathrm (d \ mathbf (x)) = a ^ (2) <\ infty) </Dd> <P> for some t, then ψ = ψ / a is called the normalized wavefunction . Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle . Hence, at a given time t, ρ (x) = ψ (x, t) is the probability density function of the particle's position . Thus the probability that the particle is in the volume V at t is </P>

What is the amplitude of an electron wave