91Ó°ÊÓ

The Schrödinger equation, or Schrödinger's wave equation, is a partial differential equation that utilizes the wave function to describe the behavior of quantum mechanical systems. The Schrödinger equation can be used to determine the trajectory, location, and energy of these systems.

(a)

For an unstable particle

$P\left(t\right)={∫}_{∞}^{∞}{\left|\mathrm{ψ}\left(x,t\right)\right|}^{2}dx=e^{-t/z}$

…(1)

Where for a stable particle (as indicated in eq.(1.27) in the book)

$P\left(t\right)={∫}_{∞}^{∞}{\left|\mathrm{ψ}\left(x,t\right)\right|}^{2}dx=0$ …(2)

Since the potential energy for an unstable particle become $V=V_0-i\mathrm{Γ}$, Schrodinger equation become

$I\mathrm{Ä\S }\frac{∂\mathrm{ψ}}{∂T}=\frac{{\mathrm{Ä\S }}^2{∂}^2\mathrm{ψ}}{2m∂x^2}+\left(V_0-i\mathrm{Γ}\right)\mathrm{ψ}$ …(3)

Where its conjugate is

$-i\mathrm{Ä\S }\frac{∂\mathrm{ψ}}{∂t}=-\frac{{\mathrm{Ä\S }}^2}{2m}\frac{{∂}^2\mathrm{ψ}}{∂x^2}+\left(V_0+i\mathrm{Γ}\right)\mathrm{ψ}$ …(4)

Now,

$$\begin{gathered} \frac{dp}{dt}=\frac{d}{dt}{∫}_{-∞}^{∞}{\left|\mathrm{ψ}\right|}^{2}dx \\ ={∫}_{-∞}^{∞}\frac{∂}{∂t}{\mathrm{ψ}}^{*} \\ \frac{dp}{dt}={∫}_{-∞}^{∞}\left(\mathrm{ψ}\frac{∂{\mathrm{ψ}}^{*}}{∂t}+{\mathrm{ψ}}^{*}\frac{∂\mathrm{ψ}}{∂t}\right)dx \end{gathered}$$ …(5)

Substitute from Schrodinger equation and its conjugate, thus, the integrand become

$\left(\mathrm{ψ}\left[\frac{-i\mathrm{Ä\S }}{2m}\frac{{∂}^2{\mathrm{ψ}}^{*}}{∂x^2}-\frac{1}{\mathrm{Ä\S }}{V}_0{\mathrm{ψ}}^{*}\frac{\mathrm{Γ}}{\mathrm{Ä\S }}{\mathrm{ψ}}^{*}\right]+{\mathrm{ψ}}^{*}\left[\frac{-i\mathrm{Ä\S }}{2m}\frac{{∂}^2{\mathrm{ψ}}^{*}}{∂x^2}+\frac{1}{\mathrm{Ä\S }}{V}_0\mathrm{ψ}-\frac{\mathrm{Γ}}{\mathrm{Ä\S }}{\mathrm{ψ}}^{*}\right]\right)$

Therefore,

$$\begin{gathered} \frac{dp}{dt}={∫}_{∞}^{∞}\left(\frac{-i\mathrm{Ä\S }}{2m}\left[\mathrm{ψ}\frac{{∂}^2{\mathrm{ψ}}^{*}}{∂x^2}+{\mathrm{ψ}}^{*}\frac{{∂}^2\mathrm{ψ}}{∂x^2}\right]-2\frac{\mathrm{Γ}}{\mathrm{Ä\S }}\mathrm{ψ}{\mathrm{ψ}}^{*}\right)dx \\ =\frac{-i\mathrm{Ä\S }}{2m}{∫}_{-∞}^{∞}\left(\mathrm{ψ}\frac{{∂}^2{\mathrm{ψ}}^{*}}{∂x^2}+{\mathrm{ψ}}^{*}\frac{{∂}^2\mathrm{ψ}}{∂x^2}\right)dx-{∫}_{∞}^{∞}2\frac{\mathrm{Γ}}{\mathrm{Ä\S }}\mathrm{ψ}{\mathrm{ψ}}^{*}dx \\ \frac{dp}{dt}={∫}_{∞}^{∞}\frac{∂}{∂x}\left(\mathrm{ψ}\frac{∂{\mathrm{ψ}}^{*}}{∂x}+{\mathrm{ψ}}^{*}\frac{∂\mathrm{ψ}}{∂x^2}\right)dx-2\frac{\mathrm{Γ}}{\mathrm{Ä\S }}{∫}_{-∞}^{∞}{\left|\mathrm{ψ}\right|}^{2}dx \end{gathered}$$ …(6)

Where, the first integral is zero from eq.(6), and the second integral equal to the probability, so,

$\frac{dp}{dt}=-\frac{2\mathrm{Γ}}{\mathrm{Ä\S }}p$

Hence ,its proved.

(b)

If we derive eq.(1) with respect to time we can get

$$\begin{gathered} \frac{dp}{dt}=\frac{d}{dt}{e}^{-t/z} \\ =\frac{-1}{\mathrm{Ï„}}{e}^{t/z} \end{gathered}$$

So, from $\frac{dp}{dt}=-\frac{2\mathrm{Γ}}{\mathrm{Ä\S }}p$ we can get by substituting $\frac{dp}{dt}$ inside it

$$\begin{gathered} \frac{-e^{-t/z}}{\mathrm{Ï„}}=\frac{-2\mathrm{Γ}}{\mathrm{Ä\S }}{e}^{-t/z} \\ \mathrm{Ï„}=\frac{\mathrm{Ä\S }}{2\mathrm{Γ}} \end{gathered}$$

Therefore, the lifetime of the particle in terms of$\mathrm{Ï„}$is$\mathrm{Ï„}=\frac{\mathrm{Ä\S }}{2\mathrm{Γ}}$