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.

Postulate 1: Wave function#

Wave function

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

\[P(x,t)=|\psi(x,t)|^2 \,\,\,\]
\[\int |\psi(x,t)|^2 dx=1 \]

  • 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.

Postulate 2: Operators#

Operators

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

\[\hat{L} (c_1\psi_1+c_2\psi_2+...+c_n\psi_n)=c_1\hat{L} \psi_1+c_2\hat{L}\psi_2+...+c_n\hat{L}\psi_n \]

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

\[\int \psi^* (\hat{H} \phi) dx = \int \phi (\hat{H} \psi)^* dx =\int \phi \hat{H}^* \psi^* dx \]
\[\langle \phi |\hat{H} | \psi \rangle = \langle \psi |\hat{H} | \phi \rangle^{*}\]

Postulate 3: Eigenvalues#

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.

\[\hat{A}\psi_n =a_n \psi_n \]

Postulate 4: Expectations#

Expectations

  • Expectation or the average value of observable A in a state described by wave function \(\psi\) is given by expression:

\[\boxed{\langle A \rangle = \int \psi^* \hat{A}\psi dx = \langle \psi | A| \psi \rangle}\]

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

Postulate 5: Time evolution#

Time evolution

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

    \[\boxed{i\hbar \frac{\partial \psi}{\partial t}=\hat{H}\psi}\]