In Week 1, we introduced Probability Distributions (PDs).
Last week we looked at PDs in Quantum Mechanics (QM) and how to compare one against another. (Although if you read the comment on that post, you’ll know that we’re really dealing with the square roots of PDs, rather than PDs themselves. I’m not going to worry about the distinction here.)
“https://platitudes.home.blog/2024/06/08/the-remarkable-theory-of-quantum-fields/“
“https://platitudes.home.blog/2024/06/15/summary-of-quantum-mechanics/“
We need one more concept from QM before we move onto Quantum Field Theory (QFT) proper. This is the concept of “operators”. Operators act on PDs and transform them into different PDs. These operations perform simple arithmetic operations on the numbers in the PD. For example, we could define an operator that changes the |4> state into the |6> state. All it does is set the fourth position to zero and set the sixth position to 1: (0,0,0,1,0,0) -> (0,0,0,0,0,1). Call this operator “C”, for “change”. We write this operation as:
C |4> = |6>
(For the mathematically inclined. The C operator is a permutation matrix acting on the vector |4>.)
You should read this as: the operator C, acting on the state |4>, returns the state |6>. Using operators like this we can change one PD into another PD. They’re just blocks of numbers that tell us what arithmetic operations to perform on a PD to get a different PD. We’ll use operators extensively in QFT, but first, a bit more about their use in QM.
The rest of this post is optional. What follows below is some extra information about how operators are used in QM. You won’t need it for the Quantum Field theory that I’ll start with next week. But if you read it through, you’ll find out what the Schrodinger Equation is.
There is one operator that is special. We’ll call it the “Dice measurement operator” and denote it by D. This operator leaves the PDs unchanged except for a multiplying factor. We construct D such that:
D |1> = 1 |1>
D |2> = 2 |2>
…
D |6> = 6 |6>
(For those familiar with the maths – this is a diagonal matrix with the numbers 1 to 6 on the diagonal and |n> arranged as a column vector.)
Or, more compactly,
D |n> = n |n>.
This should be read as “the measurement operator D, acting on one of the probability distributions, |n>, returns the same probability distribution, |n>, multiplied by the measurement value, n”.
Now comes the interesting bit. Suppose we are playing a game where the scoring depends, not on the face value of the die, but on the square of the value. So if we throw a 3, our actual score is 3×3 = 3² = 9. We can easily construct an operator for this that acts on our PDs. Let’s call it S. It would act like this.
S |1> = 1 |1>
S |2> = 4 |2>
…
S |6> = 36 |6>
i.e. It just returns the same PD multiplied by the square of the face value. But there is another way to construct this.
Recall that
D |n> = n |n>.
Now suppose we take the output of the D operator ( n |n> ) and apply D again. n is just a number, and we can move numbers around as we please. Don’t worry if the next bit of maths is confusing, it’s the conclusion that’s important.
D (D |n> )
= D ( n |n> )
= n (D |n> )
= n ( n |n> )
= n² |n>
In other words, applying the D operator twice gives us the same value as applying the S operator once. If we write the application of D acting twice as D², then we have
D² |n> = S |n>.
Notice that the relationship netween the operators D and S, is exactly the same as the relationship between the things that they measure, the value on the die and the square of its value.
In QM, |n> isn’t the value of a die, but the position of a particle. There are many different measurement operators that can act on |n>. Two special ones measure the rate at which |n> changes with time, this returns the total energy. Call this operator E. Another measures the rate at which |n> changes with position, this returns the momentum and can be used to construct the kinetic energy of a particle. Call this latter operator K. Other operators can return the potential energy of a particle. Call this P.
Just like the two operators that we constructed for the dice, these operators obey the same underlying equation as the physical variables that they measure. Since total energy is the sum of kinetic and potential energy, we therefore get.
K |n> + P |n> = E |n>
This is the famous Schrodinger Equation (SE). A lot of QM involves solving this equation for different versions of P, subject to different start and end conditions. In many cases, we find that the measured values are constrained to form a discrete set. This is where the “quantum” in QM comes from.
The SE was, and remains, enormously successful. Modern atomic physics, physical chemistry, materials science and semiconductor physics rely heavily on it. Much of our modern technological world is only possible because of the enormous effort that has gone into finding solutions to the SE.
Yet, despite it’s undoubted importance, the SE is clearly not the end of the story. Even at relatively low energies, there are limits to the applicability of the SE. Particles decay. Other particles spontaneously appear. The SE often has difficulty modelling these phenomena.
The SE also treats matter and force particle differently. Matter particles have a PD, |n>, associated with them. Whereas forces are simply included in the “P” term of the equation. Yet we know from experiment that matter particles, such as electrons, and force particles, such as photons, can both display either particle or wave like properties.
A more general model, capable of handling the creation and annihilation of particles, and putting matter and forces on the same footing, is needed. This is where Quantum Field Theory comes in.
Platitude of the Day 