One might wonder Schrodinger's equation is in the form it is in. It is called a wave equation but we are not using the wave equation. Well, I can't claim to understand what I am about to expostulate( in other words give an intuitive understanding) but rather take the cope out way and argue with a combination of mathematical and experimental reasoning.
Our story begins with an electron going through the double slit and hitting a screen somewhere behind it. The surprising result was that an interference pattern was observed as a series of these electrons passed through one at a time. The obvious explanation is that the electron has a wave property so we quickly write down our trusty sine function( or cosine if you are inclined that way). Now the next step is to get the interference pattern through our results. So we do the naive thing and write down two sine functions differing by a phase factor and add them together . One must recall that we want an interference pattern that has a maximum at the center and decreases "sinusoidally" in amplitude as we go away from it from either side. Unfortunately when we go with our naive guess and square our wave function(addition of the sines) since E.M tells us that the intensity is proportional to wave function squared our interference is sinusoidal but does not decrease as we go away from the center. thus the interference pattern is still illusive.
In fact we realize something worse. Supposing two waves are approaching each other and let's stick to one dimension(we are modelling two electrons approaching each other). We expect a combination of destructive and constructive interference . So we re-write the sine functions in this manner i.e \( \sin (kx - \omega t) + \sin (kx + \omega t) \); the result is this \(2 \sin kx \cos \omega t \). But this is a terrifying result, at multiples of t = \( \frac{ \pi}{2 \omega} \) the particles disappear everywhere. This sort of tom-foolery is allowed for the Cheshire cat but nor for our dear electrons. So we want waves but adding sines or cosines together does not work!
Finally we remember something about Euler's identity and it being related to waves. So we decide to represent our electrons by this wave function \( e^{(ikx - i\omega t)} \). We add one coming from left \( e^{(ikx - i\omega t)} \) and one coming from the right \( e^{(-ikx - i\omega t)} \) and get \( 2 e^{(-i\omega t) } \cos \omega t\). Ah! the function is still sinusoidal and never disappears , but with a weird complex amplitude. Remember our goal is to describe the interference patterns on the screen, but these patterns have real amplitudes. Before we move on we take note that we either take the wave function \( e^{(ikx - i\omega t)} \) or \( e^{(-ikx + i\omega t)} \) but not both. Reason is if we take both and add them then we are back to cosine and sine functions which have all the wrong properties. So we shall take the former and assume the amplitude is A.
Back to the puzzling imaginary amplitude. This is no cause for concern because remember the intensity is the square of the wavefunction, which means taking the modulus of our wave function. First let's add two wave functions that will describe our electron (since we are looking for an interference pattern):\(A_1 e^{(ikx_1 - i\omega t)}+A_2e^{(ikx_2 - i\omega t)} \)
Now let's look at the modulus
\begin{align}
&\left(A_1 e^{(ikx_1 - i\omega t)}+A_2e^{(ikx_2 - i\omega t)}\right) \left(A_1 e^{(-ikx_1 + i\omega t)}+A_2e^{(-ikx_2 + i\omega t)}\right)\\
&\left(A_1 e^{ikx_1}+A_2e^{ikx_2 }\right) \left(A_1e^{-ikx_1}+A_2e^{-ikx_2 }\right)\\
&\text{a bit of algebra....}\\
&A_1^2 + A_2^2 + 2A_1A_2 \cos k(x_2 - x_1)
\end{align}
The formula above is quite astonishing, if we close slit 2 in our experiment and assume our wave function is the right one while holding firm to the stipulation that its modulus is the thing that matters(avoiding complex amplitudes) then we expect \(A_1^2\) namely just a mound of electrons in front of slit 1 and similarly \(A_2^2\) if we close slit 1 and leave slit 2 open. In both of these situations the cosine term is not there. Once we open both slits the interference term appears and we get our interference pattern we were looking for.
Notice the classical guess from E.M gives a cosine term squared for the intensity, which of course does not explain the places on the screen where there are no electrons hitting.
Our story begins with an electron going through the double slit and hitting a screen somewhere behind it. The surprising result was that an interference pattern was observed as a series of these electrons passed through one at a time. The obvious explanation is that the electron has a wave property so we quickly write down our trusty sine function( or cosine if you are inclined that way). Now the next step is to get the interference pattern through our results. So we do the naive thing and write down two sine functions differing by a phase factor and add them together . One must recall that we want an interference pattern that has a maximum at the center and decreases "sinusoidally" in amplitude as we go away from it from either side. Unfortunately when we go with our naive guess and square our wave function(addition of the sines) since E.M tells us that the intensity is proportional to wave function squared our interference is sinusoidal but does not decrease as we go away from the center. thus the interference pattern is still illusive.
In fact we realize something worse. Supposing two waves are approaching each other and let's stick to one dimension(we are modelling two electrons approaching each other). We expect a combination of destructive and constructive interference . So we re-write the sine functions in this manner i.e \( \sin (kx - \omega t) + \sin (kx + \omega t) \); the result is this \(2 \sin kx \cos \omega t \). But this is a terrifying result, at multiples of t = \( \frac{ \pi}{2 \omega} \) the particles disappear everywhere. This sort of tom-foolery is allowed for the Cheshire cat but nor for our dear electrons. So we want waves but adding sines or cosines together does not work!
Finally we remember something about Euler's identity and it being related to waves. So we decide to represent our electrons by this wave function \( e^{(ikx - i\omega t)} \). We add one coming from left \( e^{(ikx - i\omega t)} \) and one coming from the right \( e^{(-ikx - i\omega t)} \) and get \( 2 e^{(-i\omega t) } \cos \omega t\). Ah! the function is still sinusoidal and never disappears , but with a weird complex amplitude. Remember our goal is to describe the interference patterns on the screen, but these patterns have real amplitudes. Before we move on we take note that we either take the wave function \( e^{(ikx - i\omega t)} \) or \( e^{(-ikx + i\omega t)} \) but not both. Reason is if we take both and add them then we are back to cosine and sine functions which have all the wrong properties. So we shall take the former and assume the amplitude is A.
Back to the puzzling imaginary amplitude. This is no cause for concern because remember the intensity is the square of the wavefunction, which means taking the modulus of our wave function. First let's add two wave functions that will describe our electron (since we are looking for an interference pattern):\(A_1 e^{(ikx_1 - i\omega t)}+A_2e^{(ikx_2 - i\omega t)} \)
Now let's look at the modulus
\begin{align}
&\left(A_1 e^{(ikx_1 - i\omega t)}+A_2e^{(ikx_2 - i\omega t)}\right) \left(A_1 e^{(-ikx_1 + i\omega t)}+A_2e^{(-ikx_2 + i\omega t)}\right)\\
&\left(A_1 e^{ikx_1}+A_2e^{ikx_2 }\right) \left(A_1e^{-ikx_1}+A_2e^{-ikx_2 }\right)\\
&\text{a bit of algebra....}\\
&A_1^2 + A_2^2 + 2A_1A_2 \cos k(x_2 - x_1)
\end{align}
The formula above is quite astonishing, if we close slit 2 in our experiment and assume our wave function is the right one while holding firm to the stipulation that its modulus is the thing that matters(avoiding complex amplitudes) then we expect \(A_1^2\) namely just a mound of electrons in front of slit 1 and similarly \(A_2^2\) if we close slit 1 and leave slit 2 open. In both of these situations the cosine term is not there. Once we open both slits the interference term appears and we get our interference pattern we were looking for.
Notice the classical guess from E.M gives a cosine term squared for the intensity, which of course does not explain the places on the screen where there are no electrons hitting.
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