Hermitian Operators, Time-Shifting Wavefunction

It was mentioned in the previous article on Quantum Mechanics [link] that if the integral of a wavefunction over all space at one time is equal to one (thereby meaning that it is normalized and that the probabilility of the particle existing is 100%), then the wavefunction is applicable to a later time, t.

qm11

A function in the place of Ψ*Ψ is used as a probability density function, ρ(x,t). The function N(t) is the resultant probability at a given time, given that the probability was found to be equal to 1 at a given time t0. Shown below, it is proposed that for dN/dt to equal zero, the Hamiltonian must be a Hermitian operator.

qm12

A Hermitian operator would satisfy the following:

herm

Hermiticity in general may referred to as a type of conjugate form of an operator. An operator is hermitian if the hermitian conjugate is equal to itself. One may compare this relationship as to a real number whose complex conjugate is equal to itself.

herm2

Returning to the calculation of dN/dt,

qm13

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