Bleah... I'm sick today with a severe case of man-flu. I had got the kids to school, and my wife is at work... so, I have a whole morning to myself to mope around and feel sorry for myself.... until I need to get the kids from their half day at school.
One of the many things that I've been meaning to do for some time... is to go back and relearn some of the Theoretical Physics and Mathematics topics that I had some passing mastery of when I was studying at University... it was something that I was relatively decent at... not the best, but not the worst! However, these classes tended to thin out pretty rapidly as you progressed through the years... and ones that survived to the end of the undergraduate degree tended to be a bit "special"... myself excepted!
So, these two books were a bit of a constant companion during some of my early years... but it has nearly been a couple of decades since I last looked at them regularly... and unfortunately, I'm finding that many things that used to be well-oiled... are now pretty damn rusty.
So, that means starting from the start again... and going slowly, trying to recover some of my lost skill and intuition. However, I guess you have to start from somewhere... at least I'm not doing it on stage with other people looking at me!
So, here is my working to show that the normalisation of a wavefunction holds across all time if the wavefunction is normalised at any point in time (easiest to choose that as t=0).
So, the opening chapter of the Cat book is a bit of refresher on probability and how it relates to Quantum Mechanics. With the principal application being the wavefunction and how it models the behaviour of a particle.
As it is a representation of the probability of finding a particle at a certain position interval at any point in time, the wavefunction must be normalised. In other words, at any given time (t) the sum across all possible positions of the wavefunction must be unity (100%). An easier way of saying this, is that any point in time... a particle has to be somewhere (the probability of finding the particle somewhere is 100%, no more and no less).
This is relatively simple problem that has it's parallel in regular statistics.... in that scaling (multiplying) the wavefunction by a constant complex number (A) will give the desired result... with the conclusion that any wavefunction that can NOT be scaled by a constant being non-representative of a particle.
It would be really handy (not to mention, quite critical) if a wavefunction, once normalised at t=0... stays normalised given no further interactions. It would be a little strange to say the least if the wavefunction of a particle started to de-normalise (missing probabilities summed over all space (x)) as time progressed... again, with no interaction. After all, it must be somewhere!
So, given a normalised wavefunction, denoted by the Greek Letter Psi (the trident looking symbol in the equation above) which is a function of position (x) and time (t)... we want to show that the change in time (d/dt) of the infinite sum across positions is zero. Another way of saying this, is that the normalisation does NOT change with time, at any point in time (t) it is somewhere with 100% percent certainty.
Like all proofs of this nature, we need to start with one side and the work to the other side... or meet in the middle. In this particular case, it is best to start with the Left Hand Side and try to morph it into the Right Hand Side... as 0 is a pretty poor starting point as it is a touch general!
Ta da! The end! When my older daughter saw this... she told me I didn't know how to write words... I told her that it was Mathematics (like her what her Grandfather used to do...), then she told me that there were no numbers.... Sigh...
Here is the beginning assumption, that the integral of the square of the wavefunction (the square of the wavefunction is equivalent to the probability distribution of the particle) over all possible positions (from negative infinity to positive infinity) is equal to 1 (100%). Not a bad assumption, as it basically states that at any point in time, the particle must be somewhere in the Universe. In case we need to refer back to it, I will name this Equation 1.
The Left Hand Side of what we want to prove... this is our starting point, and hopefully it will end up equally zero... as that would mean that the sum of all the probabilities of the wavefunction over all positions does NOT change with time. In other words, if it starts normalised at t=0, then it stays normalised.
The first line is the folding of the time derivative into the integral. On the outside of the integral, it is a total derivative of time (d/dt) as the result of the integral will be dependent ONLY on t, as the x will have been removed due to the limits on the integral (plus and minus infinity). When the derivative moves into the integral, it becomes a partial derivative (the funny looking d/dt) as the integrand (the thing being integrated) is a function of both variables x and t.
The next line is the expansion of the probability distribution (the square of the wavefunction) in to the wavefunction (Psi) times it's complex conjugate (Psi star). The reason that the square of the wavefunction is the result of the wavefunction times it's complex conjugate is that you need to remove any imaginary components of the complex wavefunction, as the probability distribution (what the square represents in reality) can ONLY be real.
The last line is a simple distribution of the partial derivative (the funny d/dt) according to the product rule.
At this stage, we stop to name this last line (and the associated LHS) Equation 2 for easy reference.... we are temporarily stuck!... but remember this point, we are coming back later once we find the right tool to crack this nut!
At this stage, we will need something more useful to substitute for the partial derivative (with respect to time) of the wavefunction and it's complex pair. Preferably something that deals in the derivatives with respect to position (x) instead of time, seeing as the integral is in terms of x....
So, we go and grab Schrodinger's Equation, which is expressed as this first line in the photo above.
The second line is a simple division of of both sides by ih(bar)... remembering from our early algebra days, that whatever you do to one side you have to do to the other (balanced buckets!). This does give us our desired partial derivative on the LHS of this equation... with no time derivatives on the other side, in fact, only position ones!
The rest of this photo describes the cleaning up of the RHS. Third line is the distribution of the multiplcation (1/ih(bar)) into the two other terms.
The fourth line is an application of the convention for simplicity and clarity that complex numbers (i) can not be at the bottom of a fraction (denominator). Thus, multiplying the top through by the identity -i^2 (which is a fancy way of saying 1) gives us enough cancellations to result in the 5th line.
At this point, we stop to name this part Equation 3.
A similar method applies for the complex conjugate of the wavefuntion (Psi star), which results in the above equation... cunningly labelled Equation 4.
Remember equation 2, which is where we left off when we got stuck? Just for east reference... here was the nut!
Nut... meet tool. We now have the ability to replace the partial time derivatives with Equations 3 and 4 (always reference your equations!)... resulting in this mess on the RHS. The first line is the direct substitution of Equations 3 and 4... and occupies two physical paper lines. I don't miss this part of things....
The second line is a tidying up of the first line. The 2nd term and 4th term cancel each other exactly... so they fall away... and for easy reference, I have factored out the constant ih(bar)/2m from the remaining terms.
Third line, the constant falls through the integral (it is not affected directly... just a scaling factor), a partial derivative operator (funny d/dx) is factored out from the existing second derivatives.
The integral, is known as the anti-derivative... as it is the reverse function of the derivative... just like plus/minus... multiply/divide... square/square root.
Thus, the integral of the partial derivative, both with respect to position (x), results in no change... aside from the application of the limits (x = +/- infinity)
Here we stop to interpret the results!
Now, by definition... the wavefunction of a particle that is normalisable (and remember, this is a normalisable wavefunction... that was our initial assumption at t=0) will tend to zero when x approaches infinity. If it does NOT approach zero, then the area under the curve (the integral) will be unbounded... and thus, it would be impossible to normalise it in the first place!
Thus, both the Wavefunction (Psi) and it's complex conjugate (Psi star) will be zero when the limits of plus/minus infinity are applied... and thus the whole thing reduces to zero.
QED!
Thus, it can be said that the LHS of the above photo, is equal to zero... which is what we set out to prove at the start. Physically speaking, the interpretation of this is that if the wavefunction is normalised at t=0, it will STAY normalised.
Well, this is hopefully the first of many forays back into the textbooks of my youth... I have quite a few things that I want to revisit... but unfortunately, much of it requires rebuilding the basic building blocks again... as decades of non-use have left my chops a little rusty!
It is nice to be finally delving into my books again... I had them shipped over from Australia a few years ago... but parenting leaves very little time to do these things. Hopefully, it will pique the interest of my young ones....
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