Friday, February 23, 2018

3. The Hamiltonian

Bread crumbs

In physics, we speak of something called the "Hamiltonian." It is the total energy of a system.

3.1 A System
What is a system? A system is something like everything that happens in a certain place. It is everything that happens in a certain context.
  • You can have an isolated system where neither energy nor matter either comes in or goes out of that context. These are really theoretical, since there is always some energy loss from a system, no matter how isolated it may be.
  • You can have an open system where both matter and energy can come in and go out of that context.
  • You can have a closed system where matter does not come or go out of the context but energy can. 
3.2 Energy
What is energy? Energy is the ability to do work or provide heat. There are two basic categories of energy:
  • kinetic energy is energy in motion, as it were. It is energy manifesting itself somehow.
  • potential energy is energy waiting to show itself. If you hold a rock in the air or pull a swing back getting ready to swing, these are examples of energy stored up, as it were, ready to be released.
If you add up the kinetic energy and the potential energy of a closed system, that gives you the total energy of that system at that moment. In a quantum system, you add up the kinetic energies of all the particles in a system, add the potential energy in that system, and you have what is called the Hamiltonian.

H = KE + PE

3.3 Kinetic Energy
The formula for kinetic energy is E = mv2. It says that the energy of something moving is the mass of the object times the square of the velocity at which it is moving.
To use this formula, you do not need to know where it came from. But it is easy enough to derive. The amount of work done on something to move it from point A to point B is the force over that distance. We say that work equals force times distance.

W = Fd

Now the work done is the change in kinetic energy. This fact is called the "work-energy theorem" in physics. You put energy into something and you have done work on it.

Force is mass times acceleration (F = ma). That is Newton's second law. So you might say that the work done on something is Work = mad (mass times acceleration times distance). Work may be crazy too.

Acceleration can be calculated in different ways, but one of the basic velocity equations is

v22 = v12 + 2ad.

If we rearrange this formula a little, we get a = (v22 - v12)/2d . And if we multiply this by the mass to get the force and the distance to get the work, we are left with:

W = 1/2 mv22 - 1/2 mv12

In other words, the difference in kinetic energy is the amount of work done, where the kinetic energy at each point is 1/2 mv2 .

3.4 Potential Energy
Unfortunately, you can't give just one equation for potential energy. It depends on the situation. So for something you are holding in the air, the potential energy is U = mgh , where
  • U stands for potential energy.
  • m is mass.
  • g is the acceleration due to gravity near the surface of the earth (9.8 m/s2).
  • h is the height.
Of course there are other potential energy situations, where energy is just waiting to happen. There are springs, for example. There is gravitational potential energy when we are no longer strictly talking about a height above the earth.

When it comes to Schrödinger's equation, the potential energy might be due to something like a gravitational field or an electric field.

3.5 Conservation of Energy
One of the laws of physics is the conservation of energy. The amount of energy at the beginning of some sort of process will be equal to the amount of energy at the end of a process. So if you have a certain amount of potential energy and you release it, you will have the same amount of kinetic energy afterwards if there is no heat lost. If you stretch a rubber band and then let go, the amount of potential energy you "stored," so to speak, in the band, will convert into motion.

Often there is a loss of heat, so we can introduce that element into the equation as well. For example, friction can take some of the released potential energy of an object rolling down an inclined plane. So the total energy of a system becomes the amount of kinetic energy plus the amount of potential energy plus whatever amount of heat is lost.

3.6 The Momentum Form
For the purposes of developing Schrödinger's equation, we want to convert the E = mv2 form of the kinetic energy equation into a slightly different form, which relates it to momentum.

In everyday language, we think of momentum as the tendency for a body in motion to stay in motion (Newton's first law). I once broke my elbow playing basketball on an unfinished concrete surface because I had a lot of momentum and no friction to stop myself.

In more formal terms, we say that momentum, p, is equal to the mass of something times its velocity.

p = mv

If we play a little with this formula, we can get it into a form that is more helpful for our purposes. So if we square both sides we get p2 = m2v2 . That means that v2 = p2/m2 . If we then substitute this v2 back into the initial E = mv2 , we get:

KE = p2/2m

So putting it all together:

H = p2/2m + U

No comments: