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## Quantum Theory

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Quantum Theory

But if an electron acts as a wave when it is moving, WHAT IS WAVING?

When light acts as a wave when it is moving, we have identified the ELECTROMAGNETIC FIELD as waving.

But try to recall: what is the electric field? Can we directly measure it?

Quantum Theory

Recall that by definition, E = F/q. We can only determine that a field exists by measuring an electric force! We have become so used to working with the electric and magnetic fields, that we tend to take their existence for granted. They certainly are a useful construct even if they don’t exist.

Quantum Theory

We have four LAWS governing the electric and magnetic fields that as a group are called:MAXWELL’S EQUATIONS(Gauss’ Law for electric fields which is equivalent to Coulomb’s Law; Gauss’ Law for magnetic fields, Ampere’s Law which is equivalent to the magnetic force law, and Faraday’s Law). By combining these laws we can get a WAVE EQUATIONfor E&M fields, and from this wave equation we can get the speed of the E&M wave and even more (such as reflection coefficients, etc.).

Quantum Theory

But what do we do for electron waves?

What laws or new law can we find that will work for matter to give us the wealth of predictive power that MAXWELL’S EQUATIONS have given us for light?

Quantum Theory

The way you get a law is to try to explain something you already know about, and then see if you can generalize. A successful law will explain what you already know about, and predict things to look for that you may not know about. This is where the validity (or at least usefulness) of the law can be confirmed.

Quantum Theory

Schrodinger started with the idea of Conservation of Energy: KE + PE = Etotal .

He noted that there are two relations between the particle and wave ideas: E=hf and p=h/ , and both could be related to energy:

- KE = (1/2)mv2 =p2/2m, and that =h/p, so that p = h/= (h/2)*(2/) = k = p, so

KE = 2k2/2m

- Etotal = hf = (h/2)*(2f) = .

Quantum Theory

He then took a nice sine wave, (actually a cosine wave which differs from a sine wave by a phase of 90o) and called whatever was waving, :

(x,t) = A cos(kx-t) = Real part of Aei(kx-t).

He noted that both k and were in the exponent, and could be obtained by differentiating. So he tried operators:

Quantum Theory

(x,t) = A cos(kx-t) = Aei(kx-t).

pop =-i[d/dx]= -i[ikAei(kx-t)] = k

= (h/2)*(2/)* = (h/ = p .

Similary:

Eop= i[d/dt]= i[-iAei(kx-t)] =

= (h/2)*(2f)* = (hf = E .

Quantum Theory

Conservation of Energy: KE + PE = Etotal

becomes with the use of the momentum and energy operators:

-(2/2m)*(d2/dx2) + PE* = i(d/dt)

which is called SCHRODINGER’S EQUATION. If it works for more than the free electron (where PE=0), then we can call it a LAW.

Quantum Theory

What is waving here?

What do we call ? thewavefunction

Schrodinger’s Equation allows us to solve for the wavefunction. The operators then allow us to find out information about the electron, such as its energy and its momentum.

Quantum Theory

To get a better handle on , let’s consider light: how did the E&M wave relate to the photon?

Quantum Theory

The photon was the basic unit of energy for the light. How did the field strength relate to the energy?

[Recall energy in capacitor,Energy = (1/2)CV2, where Efield = V/d, andfor parallel plates C// = KoA/d, so:

Energy = (1/2)CV2 =(1/2)*(KoA/d)*(Efieldd)2

= KoEfield2 * Vol, or Energy Efield2.]

The energy in the wave depended on the field strength squared.

Quantum Theory

Since Energy is proportional to field strength squared, AND energy is proportional to the number of photons, THEN that implies that the number of photons is proportional to the square of the field strength.

This then can be interpreted to mean that the square of the field strength at a location is related to the probability of finding a photon at that location.

Quantum Theory

In the same way, the square of the wavefunction is related to the probability of finding the electron!

Since the wavefunction is a function of both x and t, the probability of finding the electron is also a function of x and t!

Prob(x,t) = (x,t)2

Quantum Theory

Different situations for the electron, like being in the hydrogen atom, will show up in Schrodinger’s Equation in the potential energy (PE) part.

Different PE functions (like PE = -ke2/r for the hydrogen atom) will cause the solution of Schrodinger’s equation to be different, just like different PE functions in the normal Conservation of Energy will cause different speeds to result for the particles.

Quantum Theory

When we solve the Schrodinger Equation using the potential energy for an electron around a proton (hydrogen atom), we get a solution that gives us three quantum numbers: one for energy (n), one for angular momentum (L), and one for the z component of angular momentum (mL). Thus the normal quantum numbers come out of the theory instead of being put into the theory as Bohr had to do. There is a fourth quantum number (recall your chemistry), but we need a relativistic solution to the problem to get that fourth quantum number (spin).

Extending the Theory

How do we extend the quantum theory to systems beyond the hydrogen atom?

For systems of 2 electrons, we simply have a that depends on time and the coordinates of each of the two electrons:

(x1,y1,z1,x2,y2,z2,t)

and the Schrodinger’s equation has two kinetic energies instead of one.

Extending the Theory

It turns out that the Schodinger’s Equation can be separated (a mathematical property):

= Xa(x1,y1,z1) * Xb(x2,y2,z2) * T(t) .

This is like having electron one in state a, and having electron two in state b. Note that each state has its own particular set of quantum numbers.

Extending the Theory

However, from the Heisenberg Uncertainty Principle (i.e., from wave/particle duality), we are not really sure which electron is electron number #1 and which is number #2. This means that the wavefunction must also reflect this uncertainty.

Extending the Theory

There are two ways of making the wavefunction reflect the indistinguishability of the two electrons:

sym = [Xa(r1)*Xb(r2) + Xb(r1)*Xa(r2) ]* T(t)

and

anti = [Xa(r1)*Xb(r2) - Xb(r1)*Xa(r2) ]* T(t) .

[We don’t have to worry about being negative, since the probability (which must be positive) depends on 2 .]

Extending the Theory

sym = [Xa(r1)*Xb(r2) + Xb(r1)*Xa(r2) ]* T(t)

anti = [Xa(r1)*Xb(r2) - Xb(r1)*Xa(r2) ]* T(t) .

Which (if either) possibility agrees with experiment?

It turns out that some particles are explained nicely by the symmetric, and some are explained by the antisymmetric.

BOSONS

sym = [Xa(r1)*Xb(r2) + Xb(r1)*Xa(r2) ]* T(t)

Those particles that work with the symmetric form are called BOSONS. All of these particles have integer spin as well. Note that if boson 1 and boson 2 both have the same state (a=b), then > 0. This means that both particles CAN be in the same state at the same location at the same time.

FERMIONS

anti = [Xa(r1)*Xb(r2) - Xb(r1)*Xa(r2) ]* T(t)

Those particles that work with the anti-symmetric wavefunction are called FERMIONS. All of these particles have half-integer spin. Note that if fermion 1 and fermion 2 both have the same state, (a=b), then = 0. This means that both particles can NOT be in the same state at the same location at the same time.

Extending the Theory

BOSONS. Photons and alpha particles (2 neutrons + 2 protons) are bosons. These particles can be in the same location with the same energy state at the same time.

This occurs in a laser beam, where all the photons are at the same energy (monochromatic).

Pauli Exclusion Principle

FERMIONS. Electrons, protons and neutrons are fermions. These particles can NOT be in the same location with the same energy state at the same time.

This means that two electrons going around the same nucleus can NOT both be in the exact same state at the same time! This is known as the Pauli Exclusion Principle!

Pauli Exclusion Principle

Since no two electrons can be in the same energy state in the same atom at the same time, chemistry is possible (and so is biology, psychology, sociology, politics, and religion)!

Thus, the possibility of chemistry is explained by the wave/particle duality of light and matter with electrons acting as fermions!

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