postulates-of-quantum-mechanics

Postulate #1 = To an ensemble of physical systems one can, in certain circumstances associate a wave function or state function which contains all the information that can be known about the ensemble. This function is in general complex; it can be multiplied by an arbitrary complex number without altering its physical significance

Postulate #2 = Superposition principle

$$\psi =c_1{\psi }_1+c_2{\psi }_2$$

This is from the inference pattern observed from the double slit experiment

Postulate #3 = every dynamic variable is associated with a linear operator

• dynamic variables are measurable (observables)¹
• observables exhibit wave like functions until a measurement is taken and causes the wave function to collapse

Postulate #4 = the only result of a precise measurement of a dynamic variable A is one the eigenvalues $a_n$ of the linear operator $\hat{A}$ associated with A²

• the set of eigenvalues of A = spectrum of A
• the spectrum can be discrete or continuous
• the spectrum of an operator representing a dynamic variable must be real because the results of the measurement are real = these operators are called Hermitian operators

Postulate #5 = if a series of measurements are made of a dynamic variable A in an ensemble of systems described by a wave function $\psi$, then the expectation value, or average value, of the dynamic variable is

$$<A>=\frac{⟨\psi |A|\psi ⟩}{⟨\psi |\psi ⟩}=\frac{\int dr\psi (r{)}^{\ast }A\psi (r)}{\int dr\psi (r{)}^{\ast }\psi (r)}$$

For a general case,

$$<A>=⟨\psi |A|\psi ⟩=\int d\alpha ⟨\psi |\alpha ⟩⟨\psi |A|\psi ⟩=\int d\alpha a(\alpha )|c(\alpha ){|}^2$$

Postulate #6 = A wave functon representing any dynamic state can be expressed as a linear combination of the eigenfunctions of A, where A is the operator associated with the dynamic variable

In other words, for a continuous case =

$$\int d\alpha |\alpha ⟩⟨\alpha |=I$$

where $\{|\alpha ⟩\}$ are the set of eigenfunctions of A

Postulate #7 = the time evolution of the wavefunction of a system is determined by the time dependent Schrodinger equation

$$i\hslash \frac{\partial \psi (r,t)}{\partial t}=\mathbb{H}\psi (r,t)$$

where $\mathbb{H}$ is the Hamiltonian (or total energy) operator

$$\mathbb{H}=-\frac{{\hslash }^2}{2m}{\nabla }^2+V(r,t)$$

Time independent Schrodinger equation = when $V(r,t)$ is independent of time, the Schrodinger equation reduces to the time independent Schrodinger equation

$$\mathbb{H}{\alpha }_n(r)=E{\alpha }_n(r)or-\frac{{\hslash }^2}{2m}{\nabla }^2{\alpha }_n(r)+V(r){\alpha }_n(r)=E{\alpha }_n(r)$$

Footnotes

1. see Operators ↩

2. see Eigenvalue equations ↩