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30.8 Quantum Numbers and Rules

- The names and symbols that are given to the physical characteristics that are quantized are very important.
- This section covers some of the more important quantum numbers and rules, which apply in chemistry, material science, and far beyond the realm of atomic physics, where they were first discovered.
- We see how physics makes discoveries that allow other fields to grow.

- The energy states of bound systems are quantized because the particle wavelength can fit into the bounds of the system.
- The allowed energies are expressed as, where, for the hydrogen atom.
- The basic states of a system are labeled by the principal quantum number.
- The lowest-energy state has the first excited state.

- The energy of the system, such as the hydrogen atom, can be expressed as some function of, as can other characteristics.

- It is now known that the magnitude of angular momentum is quantized; it was first recognized by Bohr in relation to the hydrogen atom.

- There is a rule for in atoms.
- The value can be anything from zero to.
- If that is the case, then it can be 0, 1, 2, or 3.

- It can only be zero.
- The ground-state momentum for hydrogen is actually zero.
- The picture of a circular circle is not valid.
- The electron is near the nucleus.
- The uncertainty principle explains why the electron doesn't stay in the nucleus.

- The first excited state of hydrogen can be either 0 or 1, according to the rule.

- It is more convenient to state the value of, rather than calculating it.

- It is very easy to state.

- The direction of the momentum is quantized.
- This is true in all circumstances.
- The component of angular momentum along one direction in space can only have certain values.
- The direction in space is related to the direction of the magnetic field.
- This is an aspect of perception.
- Magnetic force has no meaning if there is nothing that varies with direction.

- The rule in parentheses is that the values can range from one step to the next.
- The five values are -2, -1, 0, 1, and 2.
- Each corresponds to a different energy in the presence of a magnetic field, so that they are related to the splitting of lines into parts.

- The component of a given momentum can only have certain values, which are shown here.
- The direction can only have certain angles relative to the - axis.

- The vectors are represented in Figure 30.54 with arrows pointing in the correct direction.
- The angle of interest is determined by the ratio of to to.

- We are given, so that can be either 0 or -1.
- The value was given by.

- The figure is consistent with the angles.
- The angle is quantized.
- As illustrated, the angular momentum is on cones.
- This behavior is not observed on a large scale.
- Consider that the smallest angle is for the maximum value in the example.

- For large, there are many values so that all angles become possible.

- There are two more worrisome numbers.
- Both were first discovered for electrons.
- electrons and other fundamental particles have spin that is roughly the same as a planet spinning on its axis.
- Only one magnitude of spin is allowed for a given type of particle.
- Intrinsic angular momentum is quantized by itself.
- The direction of the spin is quantized as well.

- The direction of spin is quantized the same way as the direction of momentum.

- For electrons, they can only be 1/2 or 1/2.
- The difference between spin up and spin down is referred to as spin up.

- In the later chapters, we will see that spin is a characteristic of all particles.
- There are important differences between half-integral spin particles and integral spin particles.
- Particles called pions have the same properties as protons and neutrons.

- The state of a system is determined by its particular quantum numbers.
- The principal quantum number can have values for electrons in atoms.
- The values of the quantum number are limited.
- The quantum number can only have the values for a given value.
- It is always having electron spin.

- There are two values for the spin projection quantum number.

- There are hydrogen states in Figure 30.55 that correspond to different sets of quantum numbers.
- The clouds of probability are the locations of electrons that are determined by the number of times a measurement is made.
- The pattern of probability is shown in the figure.
- The clouds of probability look different from classical ones.
- The uncertainty principle makes it impossible for us and nature to know how the electron gets from one place to another.
- Nature on a small scale is very different from that on a large scale.

- The nature of these states is determined by their sets of quantum numbers.
- One of the possibilities for the second excited state is (3, 2, 1).
- The darker the color, the greater the chance of finding the electron.

- The quantum numbers discussed in this section are valid for a broad range of particles and other systems.
- Some quantum numbers, such as intrinsic spin, are related to fundamental classifications of subatomic particles, and they obey laws that will give us further insight into the substructure of matter and its interactions.

- Spin is a property of atoms.
- Spin has no classical counterpart.
- When the spin component is measured, one can get either spin up or spin down.

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