• A tiny, dense, positive nucleus is surrounded by negative electrons.

• The electrons can occupy only certain stationary states. The lowest-energy stationary state is the ground state.

• The nucleus consists of protons and neutrons—called nucleons—held together by the strong force.

  

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• Light consists of discrete, massless photons that travel in vacuum at the speed of light.

• A photon has energy $E=hf$.

• Photons are emitted and absorbed on an all-or-nothing basis when a quantum system jumps from one energy level to another.

  

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• In quantum mechanics, a stationary state is described by a wave function. The square of the wave function is the probability density for finding the particle in a specific region of space. Energy is quantized and shown on an energy-level diagram.

  

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• Quantum models we analyzed:

  • A particle in a rigid box

  • A particle in a finite well

  • A quantum harmonic oscillator

  • A covalent bond

What are the most important tools introduced in Part VIII?

Relativity

• Space and time depend on the motion of experimenters relative to events.

  • Time dilation: $\text{Δ}t=\text{Δ}\tau /\sqrt{1-{\beta }^2}$

  • Length contraction: $L=\sqrt{1-{\beta }^2}\ell$

• Mass and energy are interchangeable.

  • Total energy: $E={\gamma }_{\text{p}}m{c}^2=E_0+K$

  • Rest energy: $E_0=m{c}^2$

Light and matter

• Photon energy is $E=hf=hc/\text{λ}$.

• The de Broglie wavelength is $\text{λ}=h/mv$.

• The particle-in-a-box energy levels are

  $$E_n=n^2{\displaystyle \frac{h^2}{8m{L}^2}}$$

Quantum mechanics

• A particle is described by its wave function $\psi \left(x\right)$.

  • $P\left(x\right)=\left|\psi \left(x\right)\right|{}^2$ is the probability density for finding the particle.

  • The probability of finding a particle in the interval $x_{\text{L}}\le x\le x_{\text{R}}$ is

    $$\text{Prob}={\displaystyle {\int }_{x_{\text{L}}}^{x_{\text{R}}}{\left|\psi \left(x\right)\right|}^{2}dx}$$

• The probability of tunneling through a barrier of width $w$ is

  $$P_{\text{tunnel}}=e^{-2w/\eta }$$

  where the penetration distance is

  $$\eta =\hslash /\sqrt{2m(U_0-E)}$$

• Heisenberg uncertainty is ${\text{Δ}}_x\text{Δ}{}_{p_x}\ge h/2$.

Atoms and nuclei

• Atomic energies (except for hydrogen) depend on quantum numbers $n$ and $L$.

  • Hydrogen energies: $E_n=-13.60/n^2\text{eV}$.

  • The Pauli exclusion principle predicts the ground-state electron configuration.

  • A photon with $E_{\text{photon}}=\text{Δ}{E}_{\text{atom}}$ is absorbed or emitted in a quantum jump.

• Excited-state and unstable-nuclei populations decrease exponentially with time:

  $$N=N_0{e}^{-t\text{/}\tau }=N_0{\left(\frac{1}{2}\right)}^{t\text{/}{t}_{1\text{/}2}}$$

  $\tau$ is the lifetime and $t_{1/2}=\tau \text{ln}2$ is the half-life.

• Nuclear decay modes are:

  • Alpha: Emission of a ${}^4\text{He}$ nucleus.

  • Beta: Emission of an electron or positron.

  • Gamma: Emission of a photon.