91影视

The total energy of particle is $5{\mathrm{蟺}}^2{\mathrm{鈩�}}^2/2{\mathrm{mL}}^2$.

(a) The expression for energy of two particles in one dimension is given by,

$\mathbf{E}\mathbf{=}\frac{{\mathbf{n}}_{\mathbf{1}}^{\mathbf{2}}{\mathbf{蟺}}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}}{\mathbf{2}{\mathbf{mL}}^{\mathbf{2}}}\mathbf{+}\frac{{\mathbf{n}}_{\mathbf{2}}^{\mathbf{2}}{\mathbf{蟺}}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}}{\mathbf{2}{\mathbf{mL}}^{\mathbf{2}}}$

(b) The expression for probability density is given by,role="math" localid="1659954573251" $\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{\left[}\frac{\mathbf{1}}{\mathbf{L}}\left(\sin \left(\frac{\mathrm{蟺虫}}{L}\right)\left(\sin \frac{2\mathrm{蟺虫}}{L}\right)卤\sin \left(\frac{2\mathrm{蟺虫}}{L}\right)\left(\sin \frac{\mathrm{蟺虫}}{L}\right)\right)\mathbf{\right]}}^{\mathbf{2}}$

(c) The expression for probability density is given by,

role="math" localid="1659954661753" $\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{\left[}\frac{\mathbf{1}}{\mathbf{L}}\left(\sin \left(\frac{\mathrm{蟺虫}}{L}\right)\left(\sin \frac{2\mathrm{蟺虫}}{L}\right)卤\sin \left(\frac{2\mathrm{蟺虫}}{L}\right)\left(\sin \frac{\mathrm{蟺虫}}{L}\right)\right)\mathbf{\right]}}^{\mathbf{2}}$

(d) The expression for probability density is given by,

$\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\left[\frac{1}{L}\left(\sin \left(\frac{{\mathrm{蟺虫}}_1}{L}\right)\left(\sin \frac{2{\mathrm{蟺虫}}_2}{L}\right)卤\sin \left(\frac{2{\mathrm{蟺虫}}_1}{L}\right)\left(\sin \frac{{\mathrm{蟺虫}}_2}{L}\right)\right)\right]}^{\mathbf{2}}$

(a)

Suppose ${\mathrm{n}}_1=1$and ${\mathrm{n}}_2=2$.

The total energy is calculated as,

$\mathrm{E}=\frac{{\mathrm{n}}_1^2{\mathrm{蟺}}^2{\mathrm{鈩�}}^2}{2{\mathrm{mL}}^2}+\frac{{\mathrm{n}}_2^2{\mathrm{蟺}}^2{\mathrm{鈩�}}^2}{2{\mathrm{mL}}^2}=\frac{{\left(1\right)}^2{\mathrm{蟺}}^2{\mathrm{鈩�}}^2}{2{\mathrm{mL}}^2}+\frac{{\left(2\right)}^2{\mathrm{蟺}}^2{\mathrm{鈩�}}^2}{2{\mathrm{mL}}^2}$

(b)

The probability density for symmetric function is calculated as,

$\mathrm{P}\left(\mathrm{x}\right)={\left[\frac{1}{\mathrm{L}}\left(\sin \left(\frac{\mathrm{蟺虫}}{\mathrm{L}}\right)\left(\sin \frac{2\mathrm{蟺虫}}{\mathrm{L}}\right)+\sin \left(\frac{2\mathrm{蟺虫}}{\mathrm{L}}\right)\left(\sin \frac{\mathrm{蟺虫}}{\mathrm{L}}\right)\right)\right]}^2=\frac{4}{{\mathrm{L}}^2}{\sin }^2\left(\frac{\mathrm{蟺虫}}{\mathrm{L}}\right){\sin }^2\left(\frac{2\mathrm{蟺虫}}{\mathrm{L}}\right)$

The graph between probability density and $\mathrm{x}$for symmetric and asymmetric function is shown below,

Figure 1: Curve 1 shows the symmetric function and curve 2 shown asymmetric functions.

(c)

The probability density for anti-symmetric function is calculated as,

$\mathrm{P}\left(\mathrm{x}\right)={\left[\frac{1}{\mathrm{L}}\left(\sin \left(\frac{\mathrm{蟺虫}}{\mathrm{L}}\right)\left(\sin \frac{2\mathrm{蟺虫}}{\mathrm{L}}\right)鈭�\sin \left(\frac{2\mathrm{蟺虫}}{\mathrm{L}}\right)\left(\sin \frac{\mathrm{蟺虫}}{\mathrm{L}}\right)\right)\right]}^2=0$

The graph between probability density and $\mathrm{x}$for symmetric and asymmetric function is shown below,

Figure 2: Curve 1 shows the symmetric function and curve 2 shown asymmetric function

(d)

The probability density is calculated as,

Because of the extra minus sign in the above equation comparison between part (b) and (c) shows P(x) for the symmetric wave function along the path is equal to$\mathrm{P}\left(\mathrm{x}\right)$for anti-symmetric case in case of part (b) and (c), Corresponding to curve B in figure 1 .

Similarly for anti-symmetric wave function along the path is equal to $\mathrm{P}\left(\mathrm{x}\right)$for symmetric case in case of part (b) and (c), corresponding to curve in figure 1.