This article is just to give you an idea of the mathematical form for momentum and other things in quantum mechanics and let you know what's in store...
Read this if you don't want to read the article:
Measuring the position of the particle causes the wavefunction to collapse into the "Dirac Delta Function" (look up what this function looks like if you don't know what it is then it'll make sense) so that there is complete probability to find it at one position x.
So to add onto our little "gedonken" that we started in the annotation of the link above...
"What if we made a second measurement, immediately after the first? Would you get C again, or does the act of measurement cough up a completely NEW number? On this question, EVERYONE is in agreement...a repeated measurement MUST return the same value! (the key is to repeat it quickly!!)"
So How does the Orthodox COPENHAGEN INTERPRETATIOn account for this? The first act of measurement causes the wavefunction to collapse into the Dirac delta function...it will soon spread out again, in accordance with the time evolution governed by the Schrodinger equation, so the key is that the second measurement must be made quickly before this happens!!
"The kind of question we then ask is: if we know the initial conditions of a system, that is, we know the system at time t0, what is the dynamical evolution of this system? And we use Newton’s second law for that. In quantum mechanics we ask the same question, but the answer is tricky because position and momentum are no longer the right variables to describe [the system].""
For all those who are interested (and you should be), this is the conceptual physical description as to WHAT IS the wavefunction and WHAT IS the Schrodinger equation:
So the idea is that...in the "program" of classical mechanics, the whole point is to determine the position of the particle at any given time x(t). Once we know this, we can figure out other dynamic variables like velocity, momentum, kinetic energy or potential energy. So how do we find the position at x(t)? We use Newton's second law, F=ma, (ok fine for CONSERVATIVE systems, but fortunately these are the only kinds that appear in Quantum Mechanics on small scale :) ) along with the initial conditions of the system (position and velocity at time t = 0 (or momentum and position).
So quantum mechanics also has this same problem.... except this weird wave-particle duality of matter proposed by de Broglie kind of complicates things, so that we need a NEW mathematical formulation to answer this same question of "where is the particle, or what is the displacement of the wave at each x at every point in time t." In this case, what we're looking for is then the particle's WAVE FUNCTION, and we can find it by solving the SCHRODINGER EQUATION.
**There is an easy conceptual "proof" provided in the article, that rationalizes some of the thought behind the development of the Schrodinger equation and its origins in the classical wave equation....and if you're having trouble understanding what information it is that the wavefunction is trying to provide still, try looking at the explanation and the pictures in this article link then!!
Look up the solutions to Problem 1.5, Problem 1.7, Problem 1.9 above.
If your course uses this textbook, or you have this textbook at home...well, I guess this might just be the best day of your life! :P
Okay, don't freak out, I know this page is in Chinese, but it is just the Solutions Manual to David Griffith's "Introduction to Quantum Mechanics" 2nd edition textbook! And the Solution Manual itself is in English and Math of course hahaha!
Look up the solutions to Problem 1.5, Problem 1.7, and Problem 1.9 provided above.
...If you REALLY want other practice problems relating to this material, so you can look up their solutions in this manual and get further help, you can contact me at firstname.lastname@example.org
You should BE ABLE TO SOLVE Problems 1.5 and 1.9 in order to show you have a good understanding of the material we just covered, and to pass your test haha....because these kinds of questions are GUARANTEED to be asked in any standard course SO MAKE SURE YOU CAN DO THEM (they are probably very similar to your homework questions)!!
Problem 1.7 is a little more unconventional type of question in what it asks but it is very good to understand the solution to understand QM better :)
"But what exactly IS a wavefunction, and what does it do for you once you've got it? After all, a particle by its nature, is localized at a point, whereas the wavefunction is spread out in space (it's a function of x, for any given time t, [think of one graph as a function of x at time t=0, one at time t=1, etc.] as its name suggests)." How can such an object represent the STATE of a particle? The answer is provided by Born's STATISCAL interpretation of the wave function, which says that the square of the wavefunction gives the PROBABILITY of finding the particle at point x, at time t."
NOTE: This link will open a POWERPOINT PRESENATATION lecture on your computer. :) Don't worry about the example slides 16-18 and the time dependent Schrodinger equation stuff...although it will be interesting and useful to know even at this point if you can understand it.
IMPORTANT THING TO UNDERSTAND:
SO you gotta know that "this STATISTICAL INTERPRETATION introduces a kind of indeterminacy into quantum mechanics, for even if you know EVERYTHING the theory has to tell you about the particle (the WAVE FUNCTION....like solving x(t) in classical mechanics), you STILL cannot predict with certainty the outcome of a simple experiment to measure its position -- all quantum mechanics has to offer is STATISTICAL information about the POSSIBLE results. SHOCKING, and WEIRD, huh? Well, don't worry, this indeterminacy has been VERY disturbing to physicists now and throughout history and phliosophers and so people wonder is this REALLY how physics is or is it just a defect of the theory?"
Ok so there have been three positions that people usually take on this issue:
"Let's say a measurement of the position of the particle is conducted and you find it to be at point C. So....where WAS the particle JUST BEFORE i made the measurement?
- THE REALIST - The particle WAS at C! There is a "hidden variable" to the experimenter (the position is not actually indeterminate...just unknown to the experimenter) that is needed to determine it and the wavefunction is incomplete representation.
- THE ORTHODOX POSITION (Copenhagen/Born interpretation) - The particle wasn't really anywhere!! It was the act of measurement that FORCED the particle to take a stand (though how and why it decided on point C, we dare not ask). Observations not only DISTURB what is to be measured, they PRODUCE it...we COMPEL the particle to assume a definite position.....this thought refers to the "COLLAPSING OF A WAVE FUNCTION (more on this idea in the next link)"
THE AGNOSTIC - REFUSE TO ANSWER. I HAVE NO CLUE, why do you want to know where the particle was BEFORE the measurement anyways?"
So spoiler alert: THE ORTHODOX POSITION IS RIGHT!!! John Bell did a great experiment in 1964 showing why this HAD to be the case and there was NO HIDDEN VARIABLE, but you will only have the knowledge to analyze this experiment after completing this course so bare with me :)
** Much of this explanation was lifted from David Griffith's Introduction to Quantum Mechanics textbook 2e.
- expectation value
- standard deviation
- probability (haha)
- probability density
- continuous variable
- discrete variable
for BOTH continuous AND discrete variables!!
...Have fun :)
The equation of interest is delta x * delta p >= hbar/2. The statiscal representation for this would be something like:
standard deviation of x * standard deviation of p >= hbar/2.
This article, very simply, shows the mathematical form you would use to calculate the expectation value of a certain observable in quantum mechanics.
This is how you solve problems that you will be asked so pay attention!! ;)
Ok, your textbook probably has a better and more detailed explanation and representation of this MATH that you WILL NEED to solve problems common for this material so if you need more help with JUST the math, it may be smart to look at your textbook also :)...