91影视

Schr枚dinger's wave equation, sometimes known as the Schr枚dinger equation, is a partial differential equation that uses the wave function to explain how quantum mechanical systems behave. The trajectory, position, and energy of these systems can be determined using the Schr枚dinger equation.

(a)

Consider$E=E_0+i螕$, where$E_{掳},螕鈭�\mathrm{鈩�}$. Now, we calculate the probability density of the time-dependent wave function $\mathrm{蠄}$.

$$\begin{gathered} \left|\mathrm{蠄}{\left(\mathrm{x},\mathrm{t}\right)}^2\right|={\mathrm{蠄}}^{*}\mathrm{蠄} \\ =\left({\mathrm{蠄}}^{*}{\mathrm{e}}^{\left({\mathrm{E}}_0-\mathrm{颈螕}\right)\mathrm{t}/\mathrm{魔}}\right)\left({\mathrm{蠄别}}^{-\left({\mathrm{E}}_0-\mathrm{颈螕}\right)\mathrm{t}/\mathrm{魔}}\right) \\ ={\left|\mathrm{蠄}\right|}^2{\mathrm{e}}^{2\mathrm{螕迟}/\mathrm{魔}} \end{gathered}$$

To check the normalizability of this wave-function, we integrate over all space.

$$\begin{gathered} {鈭�}_{-鈭�}^{鈭�}\left|蠄{\left(x,t\right)}^2\right|={鈭�}_{-鈭�}^{鈭�}{\left|蠄\right|}^2{e}^{2螕t/魔}dx \\ =e^{2螕t/魔}{鈭�}_{-鈭�}^{鈭�}{\left|蠄\right|}^{2}dx \\ =e^{2螕t/魔} \end{gathered}$$

So, it is not normalized (because the energy is not real), so to make it normalized we make $\mathrm{螕}=0$, and in general,$E$ must be real for a normalized wave-function.

(b)

Eq.(2.5) is

$\frac{{魔}^2}{2m}\frac{d^2蠄}{d{x}^2}+V蠄=E蠄$

Now, if $\mathrm{蠄}$satisfy this equation, then its conjugate ${\mathrm{蠄}}^{*}$also satisfy eq.(2.5), where $\mathrm{蠄}$and ${\mathrm{蠄}}^{*}$are complex.

$\frac{{魔}^2}{2m}\frac{d^2蠄}{d{x}^2}+V{蠄}^{*}=E{蠄}^{*}$

So, any linear combination of these two function must satisfy eq.(2.5) too (i.g.,$\left(\mathrm{蠄}+{\mathrm{蠄}}^{*}\right)$and $i\left(蠄={蠄}^{*}\right)$), in general

$蠄=a_1蝇+a_{2}x$

Where $蝇$and $x$are wave-functions, and $a_1$and $a_2$are complex constants.

$$\begin{gathered} -\frac{{魔}^2}{2m}\frac{d^2蠄}{d{x}^2}+V蠄=-\frac{{魔}^2}{2m}\frac{d^2\left(a_1蝇+a_{2}x\right)}{d{x}^2}+V\left(a_1蝇+a_{2}x\right) \\ =E\left(a_1蝇+a_{2}x\right) \end{gathered}$$

$$\begin{gathered} -\frac{{魔}^2}{2m}\left(a_1\frac{d^2蝇}{d{x}^2}+a_2\frac{d^{2}x}{d{x}^2}\right)+a_{1}V蝇+a_2{V}_x=a_{1}E蝇+2Ex \\ -\frac{{魔}^2}{2m}\left(a_1\frac{d^2蝇}{d{x}^2}+a_2\frac{d^{2}x}{d{x}^2}\right)+a_{1}V蝇+a_2{V}_x=a_{1}E蝇+2Ex \\ {a}_1\left(-\frac{{魔}^2}{2m}\frac{d^2蝇}{d{x}^2}+V蝇\right)+a_2\left(-\frac{{魔}^2}{2m}\frac{d^{2}x}{d{x}^2}+Vx\right)=a_{1}E蝇+a_{2}Ex \\ -\frac{{魔}^2}{2m}\frac{d^2蠄}{d{x}^2}+V蠄=E蠄 \end{gathered}$$

So from any complex solution, we can always construct two real solutions.

(c)

For an odd time-independent wave-function $\mathrm{蠄}\left(-\mathrm{x}\right)=-\mathrm{蠄}\left(\mathrm{x}\right)$, so

$-\frac{{\mathrm{魔}}^2}{2\mathrm{m}}\frac{{\mathrm{d}}^2\mathrm{蠄}\left(\mathrm{x}\right)}{{\mathrm{dx}}^2}-\mathrm{痴蠄}\left(\mathrm{x}\right)=-\mathrm{贰蠄}\left(\mathrm{x}\right)$

Therefore,

$-\frac{{\mathrm{魔}}^2}{2\mathrm{m}}\frac{{\mathrm{d}}^2\mathrm{蠄}\left(\mathrm{x}\right)}{{\mathrm{dx}}^2}+\mathrm{痴蠄}\left(\mathrm{x}\right)=-\mathrm{贰蠄}\left(\mathrm{x}\right)$

For an even time-independent wave-function$蠄\left(-x\right)=蠄\left(x\right)$, so

$$\begin{gathered} -\frac{{\mathrm{魔}}^2}{2\mathrm{m}}\frac{{\mathrm{d}}^2\mathrm{蠄}\left(\mathrm{x}\right)}{{\mathrm{dx}}^2}+\mathrm{痴蠄}\left(-\mathrm{x}\right)=\mathrm{贰蠄}\left(-\mathrm{x}\right) \\ -\frac{{\mathrm{魔}}^2}{2\mathrm{m}}\frac{{\mathrm{d}}^2\mathrm{蠄}\left(\mathrm{x}\right)}{{\mathrm{dx}}^2}+\mathrm{痴蠄}\left(\mathrm{x}\right)=-\mathrm{贰蠄}\left(\mathrm{x}\right) \end{gathered}$$

As we have see, both (the even and odd wave-function) satisfy eq.(2.5), so any

linear combination must satisfy this equation too, where${蠄}_{even}\left(x\right)=\frac{1}{2}\left[蠄\left(x\right)+蠄\left(-x\right)\right]$, and ${蠄}_{odd}\left(x\right)=\frac{1}{2}\left[蠄\left(x\right)-蠄\left(-x\right)\right]$.

The general solution can then be built from a linear combination of even and odd functions.