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# 4. Summary

We have seen that at the microscopic level the world is very different from anything we are familiar with. Yet the world we are familiar with is built out of this strange quantum world. Quantum mechanics brings three radically different ideas to physics:

1. Some continuous quantities are quantized.  It is perhaps easiest to understand why this one is not noticeable in everyday life, as the steps between allowed values are typically quite small (i.e. water molecules vs. continuous water).
• Light could be described by “packets” of energy called photons.  The energy of a photon is $$E = hf$$, where $$h$$ is a universal constant known as Planck’s constant. A photon has momentum given by $$p = E/c$$.
• The information needed to label a state are known as the quantum numbers. In the examples of this chapter, $$n$$ is a quantum number.
• Confining a particle to a small region leads to quantization of energy. We discussed the explicit allowed energy levels (energy spectra) for three different systems. For each of these systems $$n$$ is a positive integer.
• The energy spectra of electrons bound to atoms in elements are unique to each element, this is the reason we can use absorption and emission spectra to identify materials.
• In addition to the shell number $$n$$, electrons in atoms have three additional quantum numbers, all of which concern angular momentum.  $$l$$ takes values between 0 and $$n-1$$.  It labels how much angular momentum the electron has going around the nucleus.  $$m_z$$ which lies between $$−l$$ and $$l$$ carries information about how much angular momentum is in the $$z$$-direction.  $$m_x$$ which takes one of two values, describes the electron's internal spin.
• The Pauli exclusion principle forbids any two electrons from sharing the same quantum state. Once accepted this principle tells us why the electrons fill up the states of an atom, rather than crowding into the low energy states. The distribution and range of quantum numbers is largely responsible for the structure of the periodic table.
2. The results of measurements are no longer deterministic, but instead have different probabilities.  For systems with large number of particles this becomes unimportant, as the probability of being a long way from the mean value is small. The source of the probability is not due to incomplete knowledge of the system, but is instead due to inextricable randomness in quantum systems.
• Objects (like photons or electrons) that we normally consider to be particles also behave like waves.  They have a wavelength given by $$\lambda = h/p = h/(mv)$$. Under circumstances like two-slit interference, this wave-like behavior can be observed more clearly.
• In the examples we explored, the square of the wave function $$\psi$$ gives the probability density for the particle being observed at a particular location.  The actual probability depends on how well we define our "location."
3. The act of measuring a system affects it in a substantial way.

We summarize the actual systems for which we explicitly gave the energy levels in the table below:

System Formula for Energy Levels
Particle of mass $$m$$ in an (inescapable) one-dimensional box of length $$L$$ (also called the infinite potential well) $$E_n = n^2 E_1 = n^2 h^2/(8mL^2)$$
Simple harmonic oscillator with classical frequency $$f$$ $$E_n = (n- \frac{1}{2}) hf$$
Single electron atom in an element with $$Z$$ protons $$E_n = -Z^2 (2 \pi)^2 mk^2e^4/(2h^2)$$
$$=-Z^2(13.6 \text{ eV})/n^2$$