Quantum Mechanics Made Very Very Easy:
I have always been very interested in modern physics and perhaps especially in Quantum Mechanics but couldn't seem to my head around it. Fortunately Prof. Scott Aaronson gave a series of lectures at Waterloo which really helped clear things up.
First, why is it that many people find quantum mechanics hard to understand?
There are two ways to teach quantum mechanics. The first way -- which for most physicists today is still the only way -- follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then you learn about the "blackbody paradox" and various strange experimental results, and the great crisis these things posed for physics. Next you learn a complicated patchwork of ideas that physicists invented between 1900 and 1926 to try to make the crisis go away. Then, if you're lucky, after years of study you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.
Today, in the quantum information age, the fact that all the physicists had to learn quantum this way seems increasingly humorous. For example, I've had experts in quantum field theory -- people who've spent years calculating path integrals of mind-boggling complexity -- ask me to explain the Bell inequality to them. That's like Andrew Wiles asking me to explain the Pythagorean Theorem.
As a direct result of this "QWERTY" approach to explaining quantum mechanics - which you can see reflected in almost every popular book and article, down to the present -- the subject acquired an undeserved reputation for being hard. Educated people memorized the slogans -- "light is both a wave and a particle," "the cat is neither dead nor alive until you look," "you can ask about the position or the momentum, but not both," "one particle instantly learns the spin of the other through spooky action-at-a-distance," etc. -- and also learned that they shouldn't even try to understand such things without years of painstaking work.
The second way to teach quantum mechanics leaves a blow-by-blow account of its discovery to the historians, and instead starts directly from the conceptual core -- namely, a certain generalization of probability theory to allow minus signs. Once you know what the theory is actually about, you can then sprinkle in physics to taste, and calculate the spectrum of whatever atom you want. This second approach is the one I'll be following here.
This idea that "quantum mechanics is probability theory with negative numbers" seems very useful. In this post I will try to see if I can make it as simple as possible (but no simpler!)
POSTULATE 1: STATE
EASY: Any quantum system can be described by a list of numbers called the STATE. e.g [3/5, 4/5] or [1, 0]
The sum of the squares of the numbers has to be 1, e.g (3/5)^2 + (4/5)^2 = 1
HARDER: The numbers in general can be complex.
POSTULATE 2: CHANGE
EASY: We can only change state by doing a matrix multiplication. e.g
|0 1| |3/5| = |4/5|
|1 0| |4/5| = |3/5|
HARDER: The matrix multiplication has to keep the sum of the squares equal to 1.
Which means the matrix has to be Unitary.
This postulate is essentially equivalent to Schrodinger's Wave Equation
POSTULATE 3: MEASUREMENT
EASY: When we do a measurement of the system we will get different outcomes with probability
equal to the SQUARE of the amplitudes. e.g
If we are in state [3/5, 4/5] then with probability 9/25 = 36% we get one result and with
probability 16/25 = 64% we get another result.
HARDER: This is not the actual postulate but a useful special case that conveys the main idea.
The actual postulate is the Born Rule but it is actually not too much more complicated.
The Uncertainty principle can be derived from this measurement postulate.
Pretty much all the problems, paradoxes and controversies of quantum mechanics stem from this postulate.
And thats pretty much it! All the applications of QM in physics and Computer Science can be done by "running" things on top of the basic QM postulates. This soon gets very complicated and messy.
The analogy with classical probability is clear:
Probability Distribution = Quantum State
Stochastic Matrix = Unitary Matrix
Prob(x) =State(x)^2
First, why is it that many people find quantum mechanics hard to understand?
There are two ways to teach quantum mechanics. The first way -- which for most physicists today is still the only way -- follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then you learn about the "blackbody paradox" and various strange experimental results, and the great crisis these things posed for physics. Next you learn a complicated patchwork of ideas that physicists invented between 1900 and 1926 to try to make the crisis go away. Then, if you're lucky, after years of study you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.
Today, in the quantum information age, the fact that all the physicists had to learn quantum this way seems increasingly humorous. For example, I've had experts in quantum field theory -- people who've spent years calculating path integrals of mind-boggling complexity -- ask me to explain the Bell inequality to them. That's like Andrew Wiles asking me to explain the Pythagorean Theorem.
As a direct result of this "QWERTY" approach to explaining quantum mechanics - which you can see reflected in almost every popular book and article, down to the present -- the subject acquired an undeserved reputation for being hard. Educated people memorized the slogans -- "light is both a wave and a particle," "the cat is neither dead nor alive until you look," "you can ask about the position or the momentum, but not both," "one particle instantly learns the spin of the other through spooky action-at-a-distance," etc. -- and also learned that they shouldn't even try to understand such things without years of painstaking work.
The second way to teach quantum mechanics leaves a blow-by-blow account of its discovery to the historians, and instead starts directly from the conceptual core -- namely, a certain generalization of probability theory to allow minus signs. Once you know what the theory is actually about, you can then sprinkle in physics to taste, and calculate the spectrum of whatever atom you want. This second approach is the one I'll be following here.
This idea that "quantum mechanics is probability theory with negative numbers" seems very useful. In this post I will try to see if I can make it as simple as possible (but no simpler!)
POSTULATE 1: STATE
EASY: Any quantum system can be described by a list of numbers called the STATE. e.g [3/5, 4/5] or [1, 0]
The sum of the squares of the numbers has to be 1, e.g (3/5)^2 + (4/5)^2 = 1
HARDER: The numbers in general can be complex.
POSTULATE 2: CHANGE
EASY: We can only change state by doing a matrix multiplication. e.g
|0 1| |3/5| = |4/5|
|1 0| |4/5| = |3/5|
HARDER: The matrix multiplication has to keep the sum of the squares equal to 1.
Which means the matrix has to be Unitary.
This postulate is essentially equivalent to Schrodinger's Wave Equation
POSTULATE 3: MEASUREMENT
EASY: When we do a measurement of the system we will get different outcomes with probability
equal to the SQUARE of the amplitudes. e.g
If we are in state [3/5, 4/5] then with probability 9/25 = 36% we get one result and with
probability 16/25 = 64% we get another result.
HARDER: This is not the actual postulate but a useful special case that conveys the main idea.
The actual postulate is the Born Rule but it is actually not too much more complicated.
The Uncertainty principle can be derived from this measurement postulate.
Pretty much all the problems, paradoxes and controversies of quantum mechanics stem from this postulate.
And thats pretty much it! All the applications of QM in physics and Computer Science can be done by "running" things on top of the basic QM postulates. This soon gets very complicated and messy.
The analogy with classical probability is clear:
Probability Distribution = Quantum State
Stochastic Matrix = Unitary Matrix
Prob(x) =State(x)^2