Postulates

Postulates#

What you need to know

Quantum Mechanics can be built on top of few postulates which have been learned from comparisons of theory with the experiments.
Postulates are mathematical requirements which ensure physically meaningul solutions to schrodinger equation and which provide connection between abstract concepts of operators and experimental observables.
Once we accept thses postulates Quantum Mechanics becomes a self-contained theory capable of explaining and computing all that is to know about atoms and molecules and their interactions with radiation.

Wave function

The square of wave function $| ψ (x, t) |^{2}$ is a probability distribution of finding the quantum object at an $x$ location in space and at a time $t$

P (x, t) = | ψ (x, t) |^{2}

\int | ψ (x, t) |^{2} d x = 1

In order to describe physically meaningful states of quantum objects as a probability distribution function, the wave function (and its square) must be Continuous, single-valued, differentiable and finite. Functions that do not satisfy these criteria are discarded as non-physical.

Operators

Operators in QM are Linear. Linearity follows from the nature of Schrödinger equation which is a linear differenetial equation.

\hat{L} (c_{1} ψ_{1} + c_{2} ψ_{2} + . . . + c_{n} ψ_{n}) = c_{1} \hat{L} ψ_{1} + c_{2} \hat{L} ψ_{2} + . . . + c_{n} \hat{L} ψ_{n}

Operators in QM are Hermtian. Hermitian property of operators guarantees that eigenvalues are strictly real.

\int ψ^{*} (\hat{H} ϕ) d x = \int ϕ (\hat{H} ψ)^{*} d x = \int ϕ {\hat{H}}^{*} ψ^{*} d x

⟨ ϕ | \hat{H} | ψ ⟩ = ⟨ ψ | \hat{H} | ϕ ⟩^{*}

Eigenvalues

Eigenvalues $a_{n}$ of an operator $\hat{A}$ are the only possible values of the corresponding observable that can be measured in the experiments.

\hat{A} ψ_{n} = a_{n} ψ_{n}

Expectations

Expectation or the average value of observable A in a state described by wave function $ψ$ is given by expression:

⟨ A ⟩ = \int ψ^{*} \hat{A} ψ d x = ⟨ ψ | A | ψ ⟩

Expected value predicts what would result from doing large number of experiments measuring observable and then taking average.

Time evolution

Time evolution of wave function is governed by time-dependent Schrödinger equation.

$i ℏ \frac{\partial ψ}{\partial t} = \hat{H} ψ$