Operators and Observables

Operators and Observables

Definition

Def Operator Operators are mathematical entities used to represent physical processes that result in the change of the state vector of the system, such as the evolution of these states with time.

Definition

Def Observable An observable is a physical property or physical quantity that can be measured. Observables manifest as linear Hermitian operators in the quantum state space.

Proposition

Prop Expectation Value of Operator Given some operator $\hat{A}$, the expectation value is then given by $⟨A⟩={\int }_{-\infty }^{\infty }{\Psi }^{\ast }\hat{A}\Psi \text{dd}z$

Hamiltonian Operator

The Hamiltonian operator is $\hat{H}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial z^2}+V(z)$Thus the TISE reduces to $\hat{H}{\psi }_n=E_n{\psi }_n$

Energy Eigenstate

An energy eigenstate is a quantum state where all observables (such as position, velocity, and spin) are independent of time. It is an eigenvector of the energy operator (Hamiltonian), meaning that it corresponds to a specific energy value (the eigenvalue) rather than a superposition of different energies. Formally, $\psi$ is an energy eigenstate if $H\psi =E\psi$

The Momentum Operator

Definition

Def Kinetic Energy Operator The kinetic energy operator is $\hat{T}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial z^2}$

Definition

Def Momentum Operator The momentum operator can be reduced from the kinetic energy operator: $\hat{p}=\sqrt{2m\hat{T}}=-i\hslash \nabla$

Proposition

Prop The free particle has a definite momentum. Proof Let a free particle have $\Psi (z)=A{e}^{-ikz}$. Then $\hat{P}(\Psi (z))=-\hslash k\Psi (z)$yielding a definite momentum $-\hslash k$.

Time Dependent States

Def Energy Probability Amplitudes Energy probability amplitudes are the

Proposition

Prop The mean energy can be obtained by $E={\sum }_n|c_n{|}^2{E}_n$Proof As $E=\int {\Psi }^{\ast }\hat{H}\Psi \text{dd}z$, we can easily derive the above equation.

Ehrenfest’s Theorem

Theorem

Thrm Ehrenfest’s Theorem Expectation or mean values of observables in quantum mechanics should obey the same relations as their classical counterparts. e.g. We can easily prove that $⟨p_z⟩=m\frac{\text{dd}⟨z⟩}{\text{dd}t}$: $\begin{array}{rl}m\frac{\text{dd}⟨z⟩}{\text{dd}t}& =m\frac{\text{dd}}{\text{dd}t}{\int }_{-\infty }^{\infty }{\Psi }^{\ast }z\Psi \text{dd}z\\ & =m{\int }_{-\infty }^{\infty }\frac{\partial {\Psi }^{\ast }}{\partial t}z\Psi \text{dd}z+m{\int }_{-\infty }^{\infty }{\Psi }^{\ast }z\frac{\partial \Psi }{\partial t}\text{dd}z\end{array}$ Then by TDSE

Theorem

Thrm Bohr’s Correspondence Principle Tending towards a Newtonian universe, one in which $\hslash \to 0$. Like in Relativity where the scale is the speed of light, the scale in quantum mechanics is $\hslash$, so if $\hslash \to 0$, then every length scale is large and the universe is classical.