Chapter 2: Q25P (page 77)

Check the uncertainty principle for the wave function in the equation? Equation 2.129.

Short Answer

Expert verified

Uncertainty of the wave equation $p^{2} = {(\frac{m α}{h})}^{2}$

Step by step solution

Define the Schrodinger equation

A differential equation that describes matter in quantum mechanics in terms of the wave-like properties of particles in a field. Its answer is related to a particle's probability density in space and time.

Determine the uncertainty principle

$Ψ (x) = \sqrt{\frac{mα}{h}} e^{- \frac{mαx}{h^{2}}}$ when $x \geq 0$

$Ψ (x) = \sqrt{\frac{mα}{h}} e^{\frac{mαx}{h^{2}}}$ when $x \leq 0$

$x = \int_{- \infty}^{\infty} {xΨ}^{2} dx = 0$

$x^{2} = \int_{- \infty}^{\infty} x^{2} Ψ^{2} dx = \frac{2 mα}{h^{2}} \int_{0}^{\infty} x^{2} e^{- \frac{2 mα}{h^{2}}} dx$

$x = \frac{h^{4}}{2 m^{2} α^{2}}$

$σ_{x} = \frac{h^{2}}{\sqrt{2} mα}$

$\frac{dΨ}{dx} = - \frac{mα}{h^{2}} e^{\frac{- mαx}{h^{2}}}$ for $x \geq 0$

$\frac{dΨ}{dx} = - \frac{mα}{h^{2}} e^{\frac{mαx}{h^{2}}}$ for $x \leq 0$

$\frac{dΨ}{dx} = {(\frac{\sqrt{mα}}{h})}^{3} [- θ (x) e^{\frac{- mαx}{h^{2}}} + θ (- x) e^{\frac{mαx}{h^{2}}}]$

$\frac{d^{2} Ψ}{{dx}^{2}} = {(\frac{\sqrt{mα}}{h})}^{3} [- δ (x) e^{\frac{- mαx}{h^{2}}} + \frac{mα}{h^{2}} θ (x) e^{\frac{- mαx}{h^{2}}} - δ (- x) e^{\frac{mαx}{h^{2}}} + \frac{mα}{h^{2}} θ (- x) e^{\frac{mαx}{h^{2}}}]$

Here,

$δ (- x) = δ (x)$ ,

$f (x) δ (x) = f (0) δ (x)$

$θ (- x) + θ (x) = 1$

$\frac{d^{2} Ψ}{{dx}^{2}} = {(\frac{\sqrt{mα}}{h})}^{3} [- 2 δ (x) + \frac{mα}{h^{2}} e^{\frac{- mαx}{h^{2}}}]$

$p = 0$

$p^{2} = - h^{2} \int_{- \infty}^{\infty} Ψ \frac{d^{2} Ψ}{d x^{2}} d x$

$p^{2} = - h^{2} {(\frac{\sqrt{m α}}{h})}^{3} \int_{- \infty}^{\infty} e^{- \frac{m α x}{h^{2}}} [- 2 δ (x) + \frac{m α}{h} e^{- \frac{m α x}{h^{2}}}] d x$

$p^{2} = {(\frac{m α}{h})}^{2} [2 - 2 \frac{m α}{h^{2}} \int_{0}^{\infty} e^{- \frac{2 m α}{h^{2}}} d x]$

$\begin{matrix} p^{2} = {(\frac{m α}{h})}^{2} [1 - \frac{m α}{h^{2}} \frac{h^{2}}{2 m α}] \\ = {(\frac{m α}{h})}^{2} \end{matrix}$

$p^{2} = {(\frac{m α}{h})}^{2}$

$σ_{p} = \frac{m α}{h}$

$\begin{matrix} σ_{x} σ_{p} = \frac{h^{2}}{\sqrt{2} m α} \frac{m α}{h} \\ = \sqrt{2} \frac{h}{2} > \frac{h}{2} \end{matrix}$

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Check the uncertainty principle for the wave function in the equation? Equation 2.129.

Short Answer

Step by step solution

Define the Schrodinger equation

Determine the uncertainty principle

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