《大学物理》卢德鑫
NJU: University Physics II - 许望

Energy Quantization

$\nu\lambda=c$
Blackbody radiation(infrared to visible spectrum): $u=u(\nu,T)$ or $u=u(\lambda,T)$
Stefan-Boltzman Law: $J_B=\sigma T^4$
relation between density of outgoing energy flow $J_u$ and density of energy u in the cavity: $J_u=\frac{1}{4}uc$, $u(T)=\int_0^\infty(\nu,T)d\nu$
Radiation spectrun(or spectrum density): $u(\nu,T)$
mean energy $\overline{\varepsilon}$: $u(\nu)d\nu=\frac{8\pi\nu^2}{c^3}\overline{\varepsilon}d\nu$

Wien formula: $u(v,T)d\nu\sim v^3e^{\frac{-\alpha\nu}{T}}d\nu$
- $\overline{\varepsilon} = hve^{-\beta hv}$
Wien displacement law: $\frac{h\nu_m}{k_BT}=2.822$
Rayleigh-Jeans law: $u(\nu,T)d\nu = \frac{8\pi\nu^2}{c^3}k_BTd\nu$
- $\overline{\varepsilon} = 2*\frac{1}{2}k_BT$
definition: $\beta = \frac{1}{k_BT}$
(1900)Planck’s formula: $u(\nu,T)d\nu=\frac{8\pi\nu^2}{c^3}*\frac{h\nu}{e^{\beta h\nu}-1}d\nu$
- $\overline{\varepsilon} = \frac{hv}{e^{\beta h\nu}-1}$
Planck’s relation: $E=h\nu$
(1888) The photoelectric effect(visible to ultraviolet)
stopping potential $V_0=\frac{hv}{e}-\frac{\phi}{e}$: $T_{max}=eV_0$
cutoff frequency $\nu_c=\frac{\phi}{h}$
Einstein’s theory: $h\nu-\phi=T_{max}$
Planck’s formula: $u(\nu,T)d\nu$ = $h\nu dn(\nu)$
- the distribution function of number density: $dn(\nu)=\frac{8\pi\nu^2}{c^3}\frac{1}{e^{\beta h\nu}-1}d\nu$
- $n(T) = \frac{8\pi}{c^3(\beta h)^3}\Gamma(3)\zeta(3)$
light intensity: $I=J*hv$
(1923) Compton effect(from X ray to gamma ray)
$|\vec p|=\frac{E}{c}$
$\lambda’-\lambda=\frac{h}{m_0c}(1-cos\phi)$, Compton wavelength $\Lambda\equiv\frac{h}{m_0c}=2.426*10^{-12}$
inverse Compton effect
Spectroscopy
Balmer series: $\frac{1}{\lambda}=R_{\infty}(\frac{1}{2^2}-\frac{1}{n^2})$
Rydberg constant: $R_\infty = \frac{1}{91.13nm}$
$\frac{1}{\lambda} = R_\infty(\frac{1}{m^2}-\frac{1}{n^2}),n=m+1,m+2,\cdots$, n=1 Lyman series
Bohr Model
- Stationary states: in states satisfying BBohr-Sommerfeld quantization condition($mv r=n\hbar$), eletron has fixed energy and does not radiate.
- Microscopic energy conservation and quantum transition: Emission happend only when electron jumps from a state to another state of lower energy: $h\nu_{nm} = E_n - E_m$
- Correspondence principle: Quantum theory must agree with classical theory in the limit in which classical theory is known to agree with experiment.
- $r_n=\frac{4\pi\varepsilon_0\hbar^2}{m_ee^2}n^2\equiv a_0n^2$
- $E_n=-\frac{1}{4\pi\epsilon_0}\frac{e^2}{2a_0}\frac{1}{n^2}=-\frac{e_s^2}{2a_0}\frac{1}{n^2}=-\frac{13.6}{n^2}eV$
- $R_\infty=\frac{1}{4\pi\varepsilon}{\frac{e^2}{2a_0}}\frac{1}{hc}$
Moving nucleus:
- $R_m\equiv\frac{\mu}{m_e}R_\infty=\frac{1}{1+\frac{m_e}{m’}}R_\infty\equiv\frac{1}{\eta}R_\infty$
- $E_n=-\frac{1}{4\pi\varepsilon_0}\frac{e^2}{2\eta a_0}\frac{1}{n^2}$
- Rydberg equation: $\frac{c_{air}}{\lambda}=\frac{cR_\infty}{\eta}(\frac{1}{m^2}-\frac{1}{n^2})$

Wave Nature of Matter

wave-particle duality: $E=h\nu=\hbar\omega,p=\frac{hv}{c}=\frac{h}{\lambda}=\hbar k$
mass-momentum relation
- $p=\sqrt{2mE_k}$
- $E^2=p^2c^2+m^2c^4$
de Brogile wavelength: $\lambda=\frac{h}{p}=\frac{h}{\sqrt{2m_0E_k}\sqrt{1+\frac{E_k}{2m_0c^2}}}$
Relativistic relation: $cp=\sqrt{(E_k+m_0c^2)^2-m_0^2c^4}$
In extremely relativistic limit: $\lambda=\frac{1240eV\cdot nm}{E_k}$
For non-relativistic electron: $\lambda=\sqrt\frac{1.504eV}{E_k}nm$
Davisson-Germer experiment(1923)
Electron two-slit experiment(1961)
Uncertainty relation: $\Delta x\Delta p\geq\frac{1}{2}\hbar$, $\Delta E\Delta t\geq\frac{1}{2}\hbar$
Uncertainty: $\Delta x\equiv\sqrt{\langle x^2\rangle -\langle x\rangle ^2}$

$Schr\ddot{o}dinger Equation$

wave function(Max Born, statistical interpretation or Copenhagen interpretation)$\Psi$:
- $\Psi$ is probability amplitude
- $|\Psi|$ is probability density
- $|\Psi(x)|^2dx$ is the probability of finding the particle in an interval $dx$ near $x$
basic properties:
- Wave function is single-value, continuous, and finite
- Normalization $\int|\Psi|^2=1$ is preferred
- Superposition principle applies.
wave function
- 1D: $\Psi(x,t) = Ae^{i(kx-\omega t)}=Ae^{\frac{i}{\hbar}(px-Et)}$
- 3D: $\Psi(x,t) = Ae^{\frac{i}{\hbar}(\vec p\cdot\vec x-Et)}$
Schrodinger equation: $i\hbar\frac{\partial}{\partial t}\Psi=-\frac{\hbar^2}{2m}\nabla^2\Psi + U\Psi$
- $\frac{\partial}{\partial t}\Psi = -\frac{i}{\hbar}E\Psi$
- $\nabla\Psi =\frac{i}{\hbar}\vec p\Psi$
- $E=\frac{p^2}{2m}$
- (for relativistic case) Klein-Gorden equation and Dirac equation
- $f(t)=Ce^{-\frac{i}{\hbar}Et}$
flux density: $\vec J=\frac{\hbar}{2im}(\overline{\Psi}\nabla{\Psi}-\Psi\nabla\overline{\Psi})$
- $\frac{\partial}{\partial t}|\Psi|^2 \equiv - \nabla\cdot\vec J$
If potential is stationary: $\Psi(\vec r,t)=\psi(\vec r)f(t)$
- Hamiltonian operator: $\hat{H}\equiv-\frac{\hbar^2}{2m}\nabla^2+U(\vec r)$
- stationary Schrodinger equation: $\hat{H}\psi=E\psi$, which is eigenvalue equation of operator $\hat{H}$
- eigenvalue: $E$, eigenfunction: $\psi$, $|\psi_n(x)|^2dx$ is the probability of finding the particle in the interval $x$ to $x+dx$
expected value of an observable: $\langle O\rangle=\int\overline{\psi}\hat{O}\psi d\tau$
- $\langle x\rangle=\int\overline{\psi}x\psi dx$
- $\langle\vec p\rangle=\int\overline{\psi}\frac{\hbar}{i}\nabla\psi d\tau$
One dimensional potential well
- $\frac{d^2\psi}{dx^2}+k^2\psi=0$, ($k^2\equiv\frac{2mE}{\hbar^2}$)
- $\psi_n(x)=\sqrt\frac{2}{a}sin(\frac{n\pi x}{a})$ ($0<x<a$)
- $E_n=\frac{\hbar^2\pi^2}{2ma^2}n^2$
Potential barrier
- solution
  - $\psi=Ae^{ik_1x}+Be^{-ik_1x},(x<0)$
  - $\psi=Ce^{ik_2x}+De^{-ik_2x},(0<x<a)$
  - $\psi=Ee^{ik_1x}+Fe^{-ik_1x},(x>a)$
  - wave number
    - $k_1^2 = \frac{2mE}{\hbar^2}$
    - $k_2^2 = \frac{2m(E-U_0)}{\hbar^2}$
    - Standard procedure:
      - set F=0
      - using matching condition at x=0, a and normalization condition to fixed coefficient
  - reflection coefficients: $R\equiv\frac{|J_R|}{|J_I|}=\frac{|B|^2}{|A|^2}$
  - transmission coefficients: $T\equiv\frac{|J_T|}{|J_I|}=\frac{|E|^2}{|A|^2}$
  - tunneling: $E<U_0(k^2<0)$ but T>0
    - $T=\frac{|J_T|k_1}{|J_I|k_3}=T_0exp(-\frac{2}{\hbar}\int^{x_2}{x_1}\sqrt{2m(U(x)-E)}dx$
Potential Step
- $U(x)=U_0\theta(x)$
- $E>U_0$
  - $\psi = Ae^{ikx} + Be^{-ikx} (x<0)$
  - $\psi = Ce^{ik_1x} (x>0)$
- $E<U_0$
  - $\psi = Ae^{ikx} + Be^{-ikx} (x<0)$
  - $\psi = De^{-\kappa x} (x>0)$
Simple harmonic oscillator
- $U=\frac{1}{2}kx^2$
- $E_0=\frac{1}{2}\hbar\omega_0$
- $\psi_0=Ae^{-\frac{1}{2}\alpha^2x^2}$

Atoms

number of proton: Z
number of neutron: N
Hamiltonian for hydrogen-like atom: $\hat{H}=-\frac{\hbar^2}{2\mu}\nabla^2-\frac{Ze_s^2}{r}$, $e_s^2\equiv\frac{1}{4\pi\epsilon_0}e^2$
In sperical coordinate system: $\nabla^2=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial}{\partial r})+\frac{1}{r^2}[\frac{1}{sin\theta}\frac{\partial}{\partial \theta}(sin\theta\frac{\partial}{\partial \theta})+\frac{1}{sin^2\theta}\frac{\partial^2}{\partial \varphi^2}]$
angular momentum operator: $-\frac{\hat{L}}{\hbar^2}\equiv\frac{1}{sin\theta}\frac{\partial}{\partial \theta}(sin\theta\frac{\partial}{\partial \theta})+\frac{1}{sin^2\theta}\frac{\partial^2}{\partial \varphi^2}$
- eigenvalue and eigenfunction: $\hat{L}^2Y_{lm}(\theta,\varphi)=l(l+1)\hbar^2Y_{lm}(\theta,\varphi)$, where l is integer and $Y_{lm}$ is spherical harmonics, subscript $|m|\leq l$
- $L^2=l(l+1)\hbar^2$
- $L_z=m\hbar$
Schordinger equation: $-\frac{\hbar^2}{2\mu}[\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial}{\partial r})-\frac{1}{r^2}\frac{\hat{L}}{\hbar^2}\psi-\frac{Ze_s^2}{r}\psi=E\psi$
- $\phi(r,\theta,\varphi)=R(r)Y(\theta,\varphi)$
- Radial
- eigenfunction $R_{nl}(r)$
- eigenvalues(Columbb potential well): $E_n=-\frac{\mu Z^2e_s^4}{2\hbar^2}\frac{1}{n^2}$
- $\phi_{nlm}=R_{nl}(r)Y_{lm}(\theta,\phi)$, $n=1,2,…$, $l=0,1,\dots,n-1$, $m=-1, -l+1, \dots, l$
  - principal quantum number: n
  - angular number: l
  - magnetic number: m
degenerate: same energy level may corresponds different quantum states of same n and different l and m
- degeneracy: $n^2$
spectroscopy:
- $l=0$: sharp series
- $l=1$: principal series
- $l=2$: diffuse series
- $l=3$: fundamental series
Selection rules(in transition): $\Delta l=\pm1$
Bohr raduis: $a_0=\frac{\hbar^2}{m_ee_s^2}$
ratio: $\eta=1+\frac{m_e}{m_p}=\frac{m_e}{\mu}$
reduced mass: $\frac{m_e+m_p}{m_pm_e}$
$E_n=-\frac{Z^2e_s^2}{2\eta a_0}\frac{1}{n^2}$
Radial probality density: $w_{nl}dr=r^2dr\int d\Omega|\phi_{nlm}|^2=R^2_{nl}(r)r^2dr$
Laguerre polynomial: $R_{nl}\sim e^{-\frac{1}{2}\rho}\rho^lL^{2l+1}_{n-1-l}(\rho),\rho\equiv\frac{2Z}{na_0}r$
- the most possible radius of largest angular quantum number: $r_{mp}=\frac{1}{Z}n^2a_0$
- $\langle r\rangle=\frac{1}{2}(3n^2-l(l+1))$
electron cloud
- $J_r=J_\theta=0$, $J_\varphi=\not 0$
Spin angular momentum: $(\frac{\vec S}{\hbar})=\pm\frac{1}{2}$
Pauli exclusion principle: No two electrons in a single atom can have the same set of quantum numbers $(n,l,m,m_s)$
exchange of two:
- anti-symmetric: Fermions(Electron)
  - $$ \psi(\vec r_1,\vec r_2) = \frac{1}{\sqrt{2!}}\left|\begin{matrix} \psi_m(\vec r_1) & \psi_m(\vec r_2)\newline \psi_n(\vec r_1) & \psi_n(\vec r_2) \end{matrix}\right|$$
- symmetric: Bosons
shell: the set of orbits with a certain value of n K,L,M
subshell:
- The capacity of each subshell is $2(2l+1)$
- The electrons will occupy the lowest energy states available
Laser: Light Amplification by the Stimulated Emission of Radiation
X rays: electromagentic radiation with wavelength smaller than nm
- characteristic spectrum
- $K_\alpha$: $L\rightarrow K$
- $K_\beta$: $M\rightarrow K$
- bremsstrahlung(breaking radiation): electrons encounter deceleration
- cutoff-wavelength: $\lambda_{min}=\frac{hc}{E_k}$

Molecules and Clusters

AO(atomic orbital): the wave function
- s orbital
  - $\psi_{100}=R_{10}Y_{00}$
  - 等概率面为同心圆，概率为 10%时，$r=2.6a_0$
- p orbital
  - $p_x=R_{21}(r)(\frac{3}{4\pi})^{\frac{1}{2}}\frac{x}{r}\leftarrow -\frac{1}{\sqrt{2}}(Y_{1,+1}-Y_{1,-1})$
  - $p_y=R_{21}(r)(\frac{3}{4\pi})^{\frac{1}{2}}\frac{y}{r}\leftarrow \frac{i}{\sqrt{2}}(Y_{1,+1}+Y_{1,-1})$
  - $p_z=R_{21}(r)(\frac{3}{4\pi})^{\frac{1}{2}}\frac{z}{r}\leftarrow Y_{1,0}$
- sp hybridization
Molecular Orbital
- Hydrogen molecule ion $H_2^+$ and s-s covalent bond
- The $H_2$ molecule and the covalent bond
- p-p covalent bond
- s-p molecular bond
- s-p directed bond
- carbon and s-p hybrid orbital
Ionic bonding
- ionization energy
- electron affinity
- Coulomb potential = ionizaiton energy - affinity
- electronegativity: Pauling scale
Vibration energy levels: $E_n=\hbar\omega(n+\frac{1}{2}),\ \omega^2=\frac{k}{\mu}$
- $\Delta n = \pm1$
Rotation energy levels: $E_L\frac{L^2}{2I}=\frac{L(L+1)\hbar^2}{2\mu R^2}$
- $\Delta L=\pm1$

Fermi and Bose Statistics

Thermal (de Broglie) wavelength: $\lambda=\frac{h}{mv_{rms}}=\frac{h}{m}\sqrt\frac{m}{3k_BT}$
- When the average spacing between particles $a>\lambda$, granular property dominant.
Degeneracy temperature: $T_0\equiv \frac{m}{3k_B}(\frac{h}{ma})^2$
- $T<T_0$: wave property is more important
Fermions: particles with odd half integar spin
- $(s=\frac{1}{2})$electrons, protons, neutrons, muons, neutrinos, quarks
- Composed of odd number of Fermions are Fermions
Bosons: particles with integer spin
- $(s=\frac{1}{2})$photons, pions, mesons, gluons
- Composite particles composed of any number of Bosons and an even number of Fermions behave as Bosons
$\frac{d^2\psi(x)}{dx^2}+k^2\psi(x)=0$
- $k^2\equiv\frac{2mE}{\hbar^2}$
- $\psi(x)=Ae^{ikx}$
one-dimensional: $E_n=\frac{\hbar^2}{2m}(\frac{\pi}{L})^2n^2\equiv E_1n^2$
- $k_n=\frac{2n\pi}{L}$
- 1 state $\leftrightarrow\frac{2\pi}{L}$ interval
- $\frac{\Delta E_n}{E_n}=\frac{2n+1}{n^2}$
energy level in 3D box: $E_{n_x,n_y,n_z}=\frac{\hbar^2}{2m}(\frac{\pi}{L})^2(n_x^2+n_y^2+n_z^2)$
- $k_i=\frac{2n_i\pi}{L}$
- 1 state $\leftrightarrow(\frac{2\pi}{L})^3$
- quasi-continuity: $\sum_i\rightarrow gV\int\frac{d^3k}{(2\pi)^3}$ or $\int g(\varepsilon)d\varepsilon$
- dengenaracy due to spin: $g=2s+1$
Density of states(DOS): $g(\varepsilon)=\frac{Vm^\frac{3}{2}}{\sqrt{2}\pi^2\hbar^3}\sqrt{\varepsilon}d\varepsilon$
- $g(\varepsilon)d\varepsilon=\frac{gV}{4\pi^2}(\frac{2m}{\hbar^2})^\frac{3}{2}\varepsilon^\frac{1}{2}d\varepsilon$ = $gV\frac{4\pi k^2}{(2\pi)^3}dk$
- energy spectrum: $\frac{\hbar^2k^2}{2m}$
Fermi-Dirac: $f_{FD}=\frac{1}{e^{\beta(E-\mu)}+1}$
Fermi energy: $T\rightarrow0,E_F\equiv\mu=\frac{\hbar^2 k_F^2}{2m}=k_BT_F$
- $E»m$: $E_F=c\hbar k_F$
Distribution at zero temperature: $f(E)=\theta(E_F-E)$
Number of state: $N=\sum{<n_i>}$
Fermi sphere(Fermi sea) in k space: $N=(2s+1)V\int\frac{d^3k}{(2\pi)^3}=\frac{V}{3\pi^2}k_F^3$
- Fermi wave number: $k_F=(3\pi^2\frac{N}{V})^\frac{1}{3}$
average energy under zero temperature: $\frac{\langle E\rangle}{N}=\frac{3}{5}E_F$
Bose-Einstein distribution: $f_{BE}=\frac{1}{e^{\beta(E_j-\mu)}-1}$
Stefan-Boltzmann Law: $\frac{U(T)}{V}=\frac{8\pi}{c^3}\frac{(k_BT)^4}{h^3}\int_0^\infty\frac{x^3dx}{e^x-1}\sim T^4$
Bose-Einstein condensation

Condensed Matter

Orientation order
- nematic phase
- cholesteric phase
- smectic phase
quasi-crystal
Penrose tilting($36^\circ,72^\circ$)
Capacity dimension (Hausdorff dimension, fractal dimensinon) $d\equiv\lim_{\epsilon\rightarrow0}\frac{\ln N(\epsilon)}{ln\frac{1}{\epsilon}}$
Kohn curve: $d=\frac{\ln 4^n}{\ln 3^n}$
Cantor set: $d=\frac{2}{3}$
Similarity dimension: When a shape is composed of n similar shape of size 1/m, then the similarity dimension is given by $\frac{\ln n}{\ln m}$
Potential between ions: $V=-\alpha\frac{1}{4\pi\epsilon_0}{e^2}{r}+\frac{A}{r^n}$
ionic cohesive energy $-V_0$: $V_0\equiv V(r_0)=-\alpha\frac{1}{4\pi\epsilon_0}\frac{e^2}{r_0}(1-\frac{1}{n})$, $r_0$ is equilibrium position.
atomic cohesive energy: $V = V_0 - E(get\ e) + E(rid\ e)$

Nuclear Physics

nucleus: ${}^A_ZX_N$
atomic number: $Z$
neutron number: $N$
mass number: $A$
$R=R_0A^{\frac{1}{3}}$
unified atomic mass: $u=m_{12_C}/12$
mass excess: $\Delta\equiv\frac{m-A}{c^2}$
Bohr magneton $\mu_N$: $\mu_p=2.793\mu_N,\mu_n=-1.912\mu_N$
Binding Energy: $B=Nm_nc^2+Zm_pc^2-mc^2$
Average binding energy: $\frac{B(Z,A)}{A}\sim 8.5MeV$
Radioactivity: $-\frac{dN}{dt}=\lambda N$, $N = N_0e^{-\lambda t}$
decay rate(activity): $R\equiv-\frac{dN}{dt}=R_0e^{-\lambda t}$, $1 Bq = 1s^{-1}$
disintegration constant: $\lambda$
Half-life: $t_{1/2}=\frac{ln2}{\lambda}$
mean life: $\overline{t} = \frac{1}{\lambda}$
short-lived: $t_{1/2}=\frac{ln2}{\ln (R_0/R(t))}t$
long-lived: $\lambda = \frac{R}{N}$
Dating: $t=\frac{\ln(R_0/R(t))}{\ln 2}t_{1/2}$
alpha decay
beta decay
gamma decay
Fission: $Q=(m-(m_1+m+2))c^2=A_1*\frac{B_1}{A_1}+A_2*\frac{B_2}{A_2}-A*\frac{B}{A}$

Leptons and Quarks

Classification of particles: photon, lepton, hadron(meson, nucleon, hyperon)
Classification
- leptons: $s=\frac{1}{2},S=0$
- quarks: $s=\frac{1}{2}$
- Field quanta
weak interation: $n+v_e\rightarrow p+e^{-1}$
Conservation
- lepton number($L_e,L_\mu,L_\tau$)
  - $e^{-1},v_e$ is 1
- baryon number($B$)
- strangeness($S$): conserve in strong interation(collision between hadrons), do not conserve in weak interation
- mass-energy: no meson acan decay into hyperon and others
- angular
- momentum
Quark
- $s=\frac{1}{2}$
- $B=\frac{1}{3}$, and for antiquark, $B=-\frac{1}{3}$
- baryon consists of $qqq$
- meson consists of $q\bar{q}$
- stange quark $S=-1$
CPT theorem
- charge conjugate invariance(not conserved in weak interaction)
- space inveriance: parity conservation(not conserved in weak interaction)
  - handedness: $sgn(ps)$
    - $\mu$,neutrino: left-handed, $-1$
      - $\bar{\mu}$: right-handed
    - axial vector in mirror changes direction(like $B$)
  - time reversal invariance