Operator

Functions that take Wave functions as inputs. An operator is generally indicated with a hat.

• Using operators: $E=K+V$ you can derive the Schrödinger Wave Equation, as it is equivalent

Measured Values

In any measurement of the observable for an operator $\hat{A}$, the only values that can ever be observed are the eigenvalues Eigenvalues satisfy

$$\hat{A}\Psi =a\Psi$$

where $a$ is a constant and the value that’s measured.

Operators

Hamiltonian

For example the Time Independent Schrödinger Wave Equation using operators:

$$\hat{H}\psi =E\psi \text{where }\hat{H}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V=\hat{K}+V$$

$\hat{H}$ is called the Hamiltonian, which is an operator yielding the total energy (kinetic + potential) Also $⟨E⟩=⟨\psi |\hat{H}|\psi ⟩$

The kinetic-energy operator

$$\hat{K}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}$$

The potential-energy operator

just multiplication by $V(x)$

The position operator

just multiplication by $x$

The momentum operator

$$\hat{p}=i\hslash \frac{\partial }{\partial x}$$

The expectation value of the momentum

$$⟨p⟩=⟨\Psi |\hat{p}|\Psi ⟩=-i\hslash {\int }_{-\infty }^{\infty }\frac{\partial \Psi (x,t)}{\partial x}dx$$

Angular Momentum Operator

$${\hat{L}}_z=\hat{x}{\hat{p}}_y-\hat{y}{\hat{p}}_x$$

Possible values of $\hat{L}$ are functions of $l$ such that $L=\sqrt{l(l+1)}\hslash$ Disagrees with Bohr Model angular momentum. Implies Electron motion is more like a vibration than it is a planetary orbit. Also note that $L_z=m_l\hslash$, and the $x$ and $y$ components are not observable, as they are not calculable, even though they exist. See Quantum Numbers. Except for $l=0$, when $L=0$ so the x and y components are 0.

Total energy operator

$$\hat{E}=i\hslash \frac{\partial }{\partial x}$$

The expectation value of the energy

$$⟨E⟩=⟨\psi |\hat{E}|\psi ⟩=i\hslash \int {\Psi }^{\ast }\frac{\partial \Psi }{\partial x}dx$$