This short series of blog entries attempts to provide some insight into some of the mathematics used in Quantum Mechanics (QM). Historically, physicists developed a collection of equations and heuristics to explain atomic and sub-atomic behaviour. This became known as Quantum Mechanics because of its tendency to predict quantised values for properties of bound systems. It quickly became clear that QM was consistent with a statistical interpretation, often referred to as the "Copenhagen Interpretation".
In these blog entries, I've attempted to turn this on its head by starting off modelling all physical observables as random variables. I've tried to show that some of the mathematical machinery of QM arises quite naturally, almost trivially, as a result. In this blog entry I want to show how we can construct one of the fundamental equations of QM: the Schrodinger Equation.
I'll start by going back to the simple example, introduced in Part I, of a six sided die being thrown. Recall that the state of the die is modelled as a column of six numbers. A definite outcome is represented by the digit 1 in the row corresponding to the face showing up after the die comes to rest, with a shorthand notation of |n> to avoid having to write the columns down in full each time. These are called the 6 "basis kets".
The possible values of a "measurement" of the die are modelled by a 6 x 6 block of numbers, that I call D.
Using the "multiply rows into columns rule" for multiplying blocks of numbers, we see that:
i.e. The measurement operator D, acting on a state of definite die value |n>, just gives us |n> back but multiplied by the measurement value itself, n. Simples!
Suppose we're playing a game with the die and the rules of the game specify that we have to take, not the face value of the die, but twice the face value. Can we construct a measurement operator for this? You won't be surprised to learn that the answer is yes. Lets call it 2D. This is just the D operator with every value multiplied by 2.
When this acts on a die in a definite state it returns the same state back again, but this time with twice the value shown.
And in general.
What about if the rules of the game specified that we take the value of the die plus two, rather than times two? Constructing this measurement operator is a little bit more subtle. Instead of adding 2 to every value in the measurement operator, we only add it to the diagonal values.
If some of these rules seem a bit arbitrary, then I encourage you to get a paper and pencil and try using the "row into columns" rule to figure out various combinations like D|2>, 2D|3>, (D+2)|4> etc.Try it, you'll soon see that
One last example of this, then I'll get to the point. Suppose the rules of our dice game required us to take the value on the dice squared? To see how to do this, we have to generalise our 2D example. We could just as easily construct a 3D example to get the value of the dice times 3 for every measurement. similarly for 4D, 5D and 6D. so we could a have a set of 6 such operators, nD. These operators behave as follows.
So we can construct a set of operators that returns n squared as a result. To a get a single operator that gives n squared have a think about what happens when we get D to act on n|n>.
The "n" in the middle is just a number that can either multiple everything on the right of it, or everything on the left of it, it doesn't matter which ("multiplication is associative" (Dn)|n> = D(n|n>) ). But multiplying everything in D by n is just our nD operator (multiplication is also commutative ab = ba).
But we know how to get n|n>, by the definition of D, D|n> = n|n>. Don't worry if you didn't follow all of that, the bottom line is:
In other words, D acting twice, D squared if you like, gives the values of the dice squared, nn.
OK, I'm done with the examples. The point of all this was to illustrate that we can take any formula involving the value of the die, n, and easily construct a corresponding measurement operator by applying the same formula to D. QM uses this technique to build equations relating different physical properties. For example, you might remember from school that:
Classically, kinetic energy is a simple function of momentum, p:
The forms of the QM total energy operator and the momentum operator are both known. Given a state |s> we can therefore construct a formula relating two ways to get the total energy from |s>.
And that's it, that's the Time Dependent Schrodinger Equation, or just "Schrodinger Equation" for short. The left hand side is an operator involving only time rates of change, and the right hand side involves only distances and their rates of change. Armed with the form of the potential energy, V, you can go off and calculate the energy levels and space probability distributions of the hydrogen atom, the harmonic oscillator and finite and infinite square wells. It's the usual starting point for most introductions to QM.
In these blog entries, I've attempted to turn this on its head by starting off modelling all physical observables as random variables. I've tried to show that some of the mathematical machinery of QM arises quite naturally, almost trivially, as a result. In this blog entry I want to show how we can construct one of the fundamental equations of QM: the Schrodinger Equation.
I'll start by going back to the simple example, introduced in Part I, of a six sided die being thrown. Recall that the state of the die is modelled as a column of six numbers. A definite outcome is represented by the digit 1 in the row corresponding to the face showing up after the die comes to rest, with a shorthand notation of |n> to avoid having to write the columns down in full each time. These are called the 6 "basis kets".
1 0 0
0 1 0
0 = |1> 0 = |2> 1 = |3> etc.
0 0 0
0 0 0
0 0 0
The possible values of a "measurement" of the die are modelled by a 6 x 6 block of numbers, that I call D.
1 0 0 0 0 0
0 2 0 0 0 0
D = 0 0 3 0 0 0
0 0 0 4 0 0
0 0 0 0 5 0
0 0 0 0 0 6
Using the "multiply rows into columns rule" for multiplying blocks of numbers, we see that:
D|n> = n|n>
i.e. The measurement operator D, acting on a state of definite die value |n>, just gives us |n> back but multiplied by the measurement value itself, n. Simples!
Suppose we're playing a game with the die and the rules of the game specify that we have to take, not the face value of the die, but twice the face value. Can we construct a measurement operator for this? You won't be surprised to learn that the answer is yes. Lets call it 2D. This is just the D operator with every value multiplied by 2.
2 0 0 0 0 0
0 4 0 0 0 0
2D = 0 0 6 0 0 0
0 0 0 8 0 0
0 0 0 0 10 0
0 0 0 0 0 12
When this acts on a die in a definite state it returns the same state back again, but this time with twice the value shown.
e.g. 2D|3> = 6|3>
And in general.
2D|n> = 2n|n>
What about if the rules of the game specified that we take the value of the die plus two, rather than times two? Constructing this measurement operator is a little bit more subtle. Instead of adding 2 to every value in the measurement operator, we only add it to the diagonal values.
3 0 0 0 0 0
0 4 0 0 0 0
(D+2) = 0 0 5 0 0 0
0 0 0 6 0 0
0 0 0 0 7 0
0 0 0 0 0 8
If some of these rules seem a bit arbitrary, then I encourage you to get a paper and pencil and try using the "row into columns" rule to figure out various combinations like D|2>, 2D|3>, (D+2)|4> etc.Try it, you'll soon see that
(D+2)|n> = (n+2)|n>
One last example of this, then I'll get to the point. Suppose the rules of our dice game required us to take the value on the dice squared? To see how to do this, we have to generalise our 2D example. We could just as easily construct a 3D example to get the value of the dice times 3 for every measurement. similarly for 4D, 5D and 6D. so we could a have a set of 6 such operators, nD. These operators behave as follows.
1D|n> = 1n|n>
2D|n> = 2n|n>
...
nD|n> = nn|n>
So we can construct a set of operators that returns n squared as a result. To a get a single operator that gives n squared have a think about what happens when we get D to act on n|n>.
D n|n>
The "n" in the middle is just a number that can either multiple everything on the right of it, or everything on the left of it, it doesn't matter which ("multiplication is associative" (Dn)|n> = D(n|n>) ). But multiplying everything in D by n is just our nD operator (multiplication is also commutative ab = ba).
D n|n> = nD |n> = nn|n>
But we know how to get n|n>, by the definition of D, D|n> = n|n>. Don't worry if you didn't follow all of that, the bottom line is:
D D|n> = nn|n>
In other words, D acting twice, D squared if you like, gives the values of the dice squared, nn.
D2 |n> = n2|n>
OK, I'm done with the examples. The point of all this was to illustrate that we can take any formula involving the value of the die, n, and easily construct a corresponding measurement operator by applying the same formula to D. QM uses this technique to build equations relating different physical properties. For example, you might remember from school that:
total energy = kinetic energy + potential energy
E = K + V
Classically, kinetic energy is a simple function of momentum, p:
K = p2/2m
The forms of the QM total energy operator and the momentum operator are both known. Given a state |s> we can therefore construct a formula relating two ways to get the total energy from |s>.
E|s> = (P2/2m)|s> + V|s>
And that's it, that's the Time Dependent Schrodinger Equation, or just "Schrodinger Equation" for short. The left hand side is an operator involving only time rates of change, and the right hand side involves only distances and their rates of change. Armed with the form of the potential energy, V, you can go off and calculate the energy levels and space probability distributions of the hydrogen atom, the harmonic oscillator and finite and infinite square wells. It's the usual starting point for most introductions to QM.
