superposition

Superposition

$${\psi }_1+{\psi }_2=?$$

Previously, we have defined

$$\psi (x)=\int A(k)\sin (kx)dk$$

So, $A(k)=?$

In general,

$$\psi (x)=\sum c_n\sin \left(\frac{n\pi x}{L}\right)$$

The previous portion($\sin \left(\frac{n\pi x}{L}\right)$) of the above formula is also the basis for our Functional Vector Space. Wave superposition gives rise through time dependence. Example:

$$\Psi (x,t)=\frac{1}{\sqrt{2}}{\Psi }_1(x,t)+\frac{1}{\sqrt{2}}{\Psi }_2(x,t)$$

Solving this by including the appropriate factor of $e^{-i{E}_{n}t/\hslash }$ we get.

$$e^{-i{E}_{n}t/\hslash }\left[\frac{1}{\sqrt{2}}{\psi }_1+\frac{e^{-i{E}_{n}t/\hslash }}{\sqrt{2}}\right]$$

And the probability density is therefore given by

$$|\Psi (x,t){|}^2={\Psi }^{\ast }\Psi =\frac{1}{2}{\psi }_1^2\frac{1}{2}{\psi }_2^2+{\psi }_2{\psi }_1\cos (E_2-E_1)t/\hslash$$

Recall:

$${\Psi }_n(x,t)=\sum c_n{\phi }_n(x)e^{-i{E}_{n}t/\hslash }$$

$t=0$

This is also the Fourier series expansion

The wave function given above is a specific example of superposition that can be generalized to the function given below.

$$\Psi (x)=c_1{\psi }_1(x)+c_2{\psi }_2(x)$$

Where $c_1$ and $c_2$ are complex numbers. We can further generalize this above function to the below function:

$$\Psi (x,t)=\sum c_n{\phi }_n(x)e^{-i{E}_{n}t/\hslash }$$

where we get

$$\phi (x)e^{-i{E}_{n}t/\hslash }={\psi }_n(x)$$

through time evolution. (adding the $e^{-i{E}_{n}t/\hslash }$ term)

Setting t=0, we can see the following

$${\phi }_m^{\ast }\Psi (x,0)=\sum C_n{\phi }_m^{\ast }{\phi }_n(x)$$ $$\int {\phi }_m^{\ast }\Psi (x,0)dx=\sum C_n\int {\phi }_m^{\ast }{\phi }_n(x)dx$$

From the integral on the the RHS of the above eq:

$$\int {\phi }_m^{\ast }{\phi }_{n}dx$$ $$\int \sin \left(\frac{m\pi x}{L}\right)\sin \left(\frac{n\pi x}{L}\right)dx$$

This yields a situation where the function is 0 where $m\ne 0$ and 1 where $m=n$. This is known as the Kronecker delta, and is denoted by

$${\delta }_{m,n}$$ $$\int {\phi }_m^{\ast }\Psi (x,0)dx=\sum C_n{\delta }_{m,n}=C_m$$

This above equation is equivalent to the Fourier coefficient.

NB: $|C_m{|}^2\sim$ probability of a particle being in the “nth” state.

For $t>0$,

$$\Psi (x,t)=\sum _n{c}_n{\phi }_n(x)e^{-i{E}_{n}t/\hslash }$$

We know that

$${\int }_0^L{\Psi }^{\ast }(x,t)\Psi (x,t)dt=1$$

Because the particle has to be somewhere. If we substitute in ${\sum }_n{c}_n^{\ast }{\phi }_n^{\ast }(x)$ for ${\Psi }^{\ast }(x,t)$ we can derive the formula above to

$$\sum _m|C_m{|}^2=1$$

where $C_m$ is the probability of finding the particle in the “m”th state.