Question

Solve Equation 3.67 for $\mathit{\Psi }\mathbf{\left(}\mathit{x}\mathbf{\right)}$. Note that $\mathbf{⟨}\mathit{x}\mathbf{⟩}$and $\mathbf{⟨}\mathbf{p}\mathbf{⟩}$are constants.

Short answer

Equation 3.67 for $\Psi \left(x\right)$ is$A{e}^{-a{(x-⟨x⟩)}^2/2\hslash }{e}^{i⟨p⟩x/\hslash }$

Step-by-step solution

The Uncertainty principle.

The uncertainty principle also called the Heisenberg uncertainty principle, or indeterminacy principle says that the position and the velocity of an object cannot be measured precisely, at the same time, even in theory.

For the position-momentum uncertainty principle becomes:

$\left(\frac{\hslash }{i}\frac{d}{dx}-⟨p⟩\right)\mathit{\Psi }\mathbf{=}\mathit{i}\mathit{a}\mathbf{(}\mathit{x}\mathbf{-}\mathbf{⟨}\mathit{x}\mathbf{⟩}\mathbf{)}\mathit{\Psi }$

Solve equation 3.67 for Ψ.

Solve equation 3.67 for $\Psi$, which is given by:

$\left(\frac{\hslash }{i}\frac{d}{dx}-⟨p⟩\right)\Psi =ia(x-⟨x⟩⟩\Psi$

Now write the equation as:

$$\begin{gathered} \frac{d\Psi }{dx}=\frac{i}{\hslash }(iax-ia⟨x⟩+⟨p⟩)\Psi \\ =\frac{a}{\hslash }\left(-x+⟨x⟩+\frac{i}{a}⟨p⟩\right)\Psi \end{gathered}$$

The above equation can be written as,

$\frac{d\Psi }{\Psi }=\frac{a}{\hslash }\left(-x+⟨x⟩+\frac{i⟨p⟩}{a}\right)dx$

Integrate both sides to get:

$\ln \Psi =-\frac{a}{2\hslash }(x-⟨x⟩{)}^2+\frac{i⟨p⟩x}{\hslash }+\ln A$

Exponentiation both sides the result is,

$\Psi =A{e}^{-a{(x-⟨x⟩)}^2/2\hslash }{e}^{i⟨p⟩x/\hslash }$

In any stationary state $(⟨p⟩=0)$, so any system in which there is a stationary state that has a gaussian wave function will have minimum position-momentum uncertainty.