# 1. Introduction

We begin our discussion of quantum mechanics with something familiar; try filling a cup with water. We know that we cannot put any more than *one *cup of water in there, right? If someone asked "how much water could go in this cup?" your answer might be “any amount of water less than *one* cup will fit in here.”

Water isn't a continuous quantity though, it's composed of individual H_{2}O molecules. A standard cup is 250 mL, which is roughly \(8.4 \times 10^{24}\) molecules of water (the exact number is not important, but you should know how to calculate it like in the example below). You cannot split a water molecule in half (and still call it water) so you cannot fit *any* amount of water into the cup: you must have 0 molecules of water, 1 molecule of water, or \(8.4 \times 10^{24}\) molecules of water in the cup. We say that the amount of water in the cup is **quantized** because the amount water in the cup can only take certain allowed values (multiples of 1 molecule).

In quantum mechanics, almost every quantity we encounter (such as energy and angular momentum) can only take certain allowed values. The word “quantum” is derived from the Latin word* quantus*, the same root word as quantity. The name "quantum mechanics" reminds us that many of the things that we discuss only appear in discrete values. So far, when objects interact by exchanging energy, momentum or angular momentum, we've assumed that we could transfer *any* amount of energy, momentum or angular momentum we wished. However quantum mechanics tells us that we can only transfer these things in descrete pieces (because they are quantized). Here we will focus on quantizing *energy *and see what the consequences are.

With the example of water, knowledge of molecules makes it easy to visualize why water is quantized. If something is quantized, it's individual pieces are called **quanta **(singular: quantum). The quantum of water is one molecule. Now consider something abstract, like energy. Is the quantum of energy always the same? The answer to this is *no: th*e *allowed energies* of a system (and therefore the energy quanta), depend on the situation. For example, an atom has different allowed energies than a mass on a spring. One of the mathematically more difficult parts of quantum mechanics is finding the allowed energies of a given system.

Example #1

How many molecules are there in a cup (250 mL) of water?

**Solution**

This solution uses the definition of moles and atomic weights as covered in Physics 7A. We know that the density of water is 1 g per cubic centimeter, or 1 g/mL. A cup of water is 250 mL, so the mass of water is

\[m_{water} = \rho V = \left( \dfrac{1 \text{ g}}{1 \text{ mL}} \right) (250 \text{ mL}) = 250 \text{ g}\]

The molecular weight of a water molecule is given roughly by: oxygen has 16 nucleons (8 protons and 8 neutrons), and hydrogen has 1 nucleon. The total molecular weight is 16 + 1 + 1 = 18. This means that one mole of water molecules has a mass of 18 grams. The number of moles \(n_{moles}\) in a cup of water is

\[250 \text{ g} = \dfrac{18 \text{ g}}{\text{mole}} \times n_{moles}\]

The cup contains 13.9 moles. Each mole, by definition, has an Avogadro's number of particles. So the total number of water molecules is

\[N = N_A \times n_{moles} = \left( 6.02 \times 10^{23} \dfrac{\text{molecules}}{\text{mole}} \right) \times (13.9 \text{ moles}) = 8.4 \times 10^{24} \text{ molecules}.\]

In the quantum mechanics unit we will discuss the allowed energies for three systems:

- A simple harmonic oscillator (a mass on a spring, atomic bonds, etc.)
- The hydrogen atom
- A particle confined in a one-dimensional box

We study these three systems to help understand how only being allowed to transfer specific amounts of energy will affect the properties of a system. After this we explore how energy is quantized in the first place, and some of the strange nature of very small objects. We take this approach for an important reason. Science is an effort to find the* simplest set of assumptions *to describe the world. Studying energy levels in this way allows us to explain wide range of phenomena, including some relevant to biology and chemistry.

In the last section, which discusses why energy is quantized, is a lot more complicated but leads to more powerful predictions. This is similar to the approach we took in Physics 7A, where we started with the three-phase model and developed the ideas of atoms and modes of to explain, at least partially, why the three-phase model worked.