91影视

wave function of a quantum-mechanical system.

Schr枚dinger equation is given by,

_{$\mathbf{-}\frac{{\mathbf{h}}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}}\frac{{\mathbf{鈭�}}^{\mathbf{2}}\mathbf{唯}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}}{\mathbf{鈭�}{\mathbf{x}}^{\mathbf{2}}}\mathbf{=}\mathbf{ih}\frac{\mathbf{鈭�}\mathbf{唯}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}}{\mathbf{鈭�}\mathbf{t}}$} 鈥�.. (1)

Where, h is Plank鈥檚 constant, m is the mass of the object, x is the position of the object, t is the time, and$\mathit{唯}$is the Wave function

Wave Group (6-21) is,

$\mathbf{唯}\left(x,t\right)\mathbf{=}{\mathbf{鈭�}}_{\mathbf{-}\mathbf{鈭�}}^{\mathbf{+}\mathbf{鈭�}}\mathbf{A}\left(k\right){\mathbf{e}}^{\mathbf{i}\left(\mathrm{kx}-\mathrm{蝇迟}\right)}\mathbf{dk}$ 鈥�.. (2)

Where, k is the wave number, A is the amplitude of the wave, and $\mathbf{蝇}$is theangular frequency.

Matter Wave dispersion relation (6-23) is,

_{$\mathbf{蝇}\left(k\right)\mathbf{=}\frac{{\mathbf{hk}}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}}$} 鈥�.. (3)

$$\begin{gathered} -\frac{{\mathrm{h}}^2}{2\mathrm{m}}\frac{{鈭�}^2}{鈭�{\mathrm{x}}^2}{鈭�}_{-鈭�}^{+鈭�}\mathrm{A}\left(\mathrm{k}\right){\mathrm{e}}^{-\mathrm{i}\left(\mathrm{kx}-\mathrm{蝇迟}\right)}\mathrm{dk}=\mathrm{ih}\frac{鈭�}{鈭�\mathrm{t}}{鈭�}_{-鈭�}^{+鈭�}\mathrm{A}\left(\mathrm{k}\right){\mathrm{e}}^{-\mathrm{i}\left(\mathrm{kx}-\mathrm{蝇迟}\right)}\mathrm{dk} \\ -\frac{{\mathrm{h}}^2}{2\mathrm{m}}{鈭�}_{-鈭�}^{+鈭�}\left(-{\mathrm{k}}^2\right)\mathrm{A}\left(\mathrm{k}\right){\mathrm{e}}^{-\mathrm{i}\left(\mathrm{kx}-\mathrm{蝇迟}\right)}\mathrm{dk}=\mathrm{ih}{鈭�}_{-鈭�}^{+鈭�}\left(-\mathrm{i蝇}\right)\mathrm{A}\left(\mathrm{k}\right){\mathrm{e}}^{-\mathrm{i}\left(\mathrm{kx}-\mathrm{蝇迟}\right)}\mathrm{dk} \end{gathered}$$

Substituting $\frac{{\mathrm{hk}}^2}{2\mathrm{m}}$for $\mathrm{蝇}\left(\mathrm{k}\right)$ in above Equation, and you have

$$\begin{gathered} -\frac{{\mathrm{h}}^2}{2\mathrm{m}}{鈭�}_{-鈭�}^{+鈭�}\left(-{\mathrm{k}}^2\right)\mathrm{A}\left(\mathrm{k}\right){\mathrm{e}}^{-\mathrm{i}\left(\mathrm{kx}-\mathrm{蝇迟}\right)}\mathrm{dk}=\mathrm{ih}{鈭�}_{-鈭�}^{+鈭�}\left(-\mathrm{i}\frac{{\mathrm{hk}}^2}{2\mathrm{m}}\right)\mathrm{A}\left(\mathrm{k}\right){\mathrm{e}}^{-\mathrm{i}\left(\mathrm{kx}-\mathrm{蝇迟}\right)}\mathrm{dk} \\ -\frac{{\mathrm{h}}^2}{2\mathrm{m}}{鈭�}_{-鈭�}^{+鈭�}\left(-{\mathrm{k}}^2\right)\mathrm{A}\left(\mathrm{k}\right){\mathrm{e}}^{-\mathrm{i}\left(\mathrm{kx}-\mathrm{蝇迟}\right)}\mathrm{dk}=-\frac{{\mathrm{h}}^2}{2\mathrm{m}}{鈭�}_{-鈭�}^{+鈭�}\left(-{\mathrm{k}}^2\right)\mathrm{A}\left(\mathrm{k}\right){\mathrm{e}}^{-\mathrm{i}\left(\mathrm{kx}-\mathrm{蝇迟}\right)}\mathrm{dk} \end{gathered}$$

You can see in the above equation, LHS = RHS.

Hence, the given general wave equation (6-21) is a solution of the free-particle Schr枚dinger equation.