91影视

It鈥檚 given that a wave function must not be diverge and must not fall to 0 or faster than$\left|{\mathrm{x}}^{-1/2}\right|$ .

A plane wave is actually a monochromatic radiation in a field of infinite extent within space travelling in a specified direction. The wave function for the plane wave is always sinusoidal and depends on time.

For the limit of zero width Dirac delta function is used. Thisfunction alwaysdiverges but the wave function for the planewave neverfallsoff.

Thus, the function of plane wave never fall off but Dirac delta function is diverged.

It鈥檚 given that: It鈥檚 given that a wave function must not be diverge and must not fall to 0 or faster than $\left|{\mathrm{x}}^{-1/2}\right|$.

Formula used:

Write the expression for the wave function of plane wave.

$\mathrm{唯}\left(\mathrm{x}\right)={\mathrm{Ae}}^{\mathrm{i}(\mathrm{kx}-\mathrm{蝇迟})}$鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�.(1)

Here, $\mathrm{唯}\left(\mathrm{x}\right)$is the wave function for the plane wave, A is the amplitude of the wave, K is the propagation constant, x is the position of the plane wave, $蝇$is the angular frequency and t is the time.

Write the expression for the complex conjugate of the wave function $\mathrm{唯}\left(\mathrm{x}\right)$.

${\mathrm{唯}}^{*}\left(\mathrm{x}\right)={\mathrm{Ae}}^{-\mathrm{i}(\mathrm{kx}-\mathrm{蝇迟})}$鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�..(2)

Now, multiply equation (1) and (2).

$\mathrm{唯}\left(\mathrm{x}\right){\mathrm{唯}}^{*}\left(\mathrm{x}\right)=\left({\mathrm{Ae}}^{\mathrm{i}(\mathrm{kx}-\mathrm{蝇迟})}\right)\left({\mathrm{Ae}}^{-\mathrm{i}(\mathrm{kx}-\mathrm{蝇迟})}\right)$

${\left|\mathrm{唯}\left(\mathrm{x}\right)\right|}^2={\mathrm{A}}^2$

Integrate both sides of the above equation.

$鈭�{\left|\mathrm{唯}\left(\mathrm{x}\right)\right|}^2\mathrm{dx}={\mathrm{A}}^2\underset{-\mathrm{b}}{\overset{\mathrm{b}}{鈭�}}\mathrm{dx}$

Substitute 1 for$鈭�{\left|\mathrm{唯}\left(\mathrm{x}\right)\right|}^2\mathrm{dx}$ due to normalization and solve the above integration.

$\begin{array}{rcl}1& =& {\mathrm{A}}^2(\mathrm{b}-(-\mathrm{b}\left)\right)\\ {\mathrm{A}}^2& =& \frac{1}{2\mathrm{b}}\\ \mathrm{A}& =& \frac{1}{\sqrt{2\mathrm{b}}}\end{array}$

Here, b is some position of the plane wave.

Thus as b approaches to infinity, A goes to zero.

A function is infinitely tall and has narrow spike like a delta function.

Write the expression for the amplitude of the function.

$\mathrm{B}=\frac{1}{\sqrt{2\mathrm{蔚}}}$鈥︹�︹�︹�︹�︹�︹�︹�︹�︹�︹�︹�︹�︹�︹��.. (3)

Here, B is the amplitude of the wave andis some postion of the wave.

Substitute 0 for $蔚$in equation (3),

$\mathrm{B}=鈭�$

Thus, as the $蔚$goes to 0 B approaches to infinity.

A plane wave is actually a monochromatic radiation in a field of infinite extent within space and travelling in a specified direction. The wave function for the plane wave is always sinusoidal and depends on time.

For the limit of zero width Dirac Delta function is used. This function is always diverges but the wave function for plane wave is never fall off.

The uncertainty of the momentum of plane wave is well defined $(\mathrm{螖辫}=0)$so the uncertainty in position is closed to infinity $(\mathrm{螖虫}=鈭�)$according to Heisenberg principle.

The uncertainty of the position of delta function is well defined $(\mathrm{螖虫}=0)$, so the uncertainty in momentum is closed to infinity$(\mathrm{螖虫}=鈭�)$according to Heisenberg principle.

Thus, the nature of the two function is complementary.