The basic description of nature is in terms of probability amplitudes or state functions ψ(r, t) such that |ψ|2 is the probability that the system is found at space-time point r, t.

Probability waves have f = E/h and λ = h/p.

Knowledge of the position r of a process destroys any information which might have existed concerning the system's momentum p. Similarly knowledge of the time at which a process occurs destroys information which might have existed concerning the total energy of the process!

A particle or system with a finite lifetime τ does not have a definite mass or energy, because ΔMc2 ∼ ℏ/τ.

A particle of mass M can appear out of nothing, if it vanishes back into nothing after a time Δt ∼ ℏ/(Mc2). Physicists call this a “virtual particle.”.

Particle in Impenetrable Box!

Note that En = n2E1.

Suppose now we have an impenetrable-walled cube with a particle inside it. Now we can have standing waves of probability in three different directions, resulting in three different quantum numbers, nx, ny and nz so that n2 in the equation above, for a particle on a line, is replaced by nx2 + ny2 + nz2.


If two processes can occur in a given experiment, and they are distinguishable, the probabilities add separately, P12 = |ψ1|2 + |ψ2|2, but if the processes are indistinguishable, the probability amplitudes interfere!    P12 = |ψ1 + ψ2|2.