Quantum Theory and Photons: MDCAT Physics notes

Black-body radiation and Planck's quantum

A black body is a perfect absorber and, at the same temperature, a perfect emitter of radiation. It gives a continuous spectrum. By Stefan's law the energy radiated per second per unit area is $E=\sigma T^4$, so doubling the temperature increases the radiated power $2^4=16$ times. The peak shifts to shorter wavelength as $T$ rises (Wien: $\lambda_{max}T=$ constant).

Max Planck (1900) explained the black-body curve by assuming energy is emitted or absorbed in discrete packets called quanta:

$$E=hf=\frac{hc}{\lambda},\qquad h=6.63\times10^{-34}\ \mathrm{J\,s}$$

Planck's constant has units J s, the same as angular momentum. A quantum of light is a photon: zero rest mass, zero charge, travels at $c$.

Photon energy across the spectrum

Since $E=hf$, higher frequency means more energetic photons. In visible light violet has the highest frequency, so a violet photon carries more energy than blue, green or red. For the same total beam energy, a violet beam therefore contains the fewest photons. Number of photons per second: $n=\dfrac{P}{hf}=\dfrac{P\lambda}{hc}$.

The photoelectric effect

Emission of electrons from a metal surface when light of suitable frequency falls on it. One photon gives all its energy to one electron.

• Work function $\phi$: the minimum energy needed to eject an electron. It depends on the metal and on the nature of its surface.
• Threshold frequency $f_o=\phi/h$: below it no electrons are emitted, however intense the light.
• Above $f_o$, the maximum kinetic energy depends on frequency, not intensity. Intensity controls only the number of electrons (the current).
• Stopping potential $V_o$: the reverse potential at which the photoelectric current becomes zero. $eV_o=K.E_{max}$, so a maximum K.E. of 5 eV needs a stopping potential of 5 V.

The photoelectric effect supports the particle nature of light; interference, diffraction and polarization support the wave nature. A photocell works on the photoelectric effect.

Photon momentum and de Broglie waves

A photon has no rest mass but carries momentum $p=\dfrac{E}{c}=\dfrac{hf}{c}=\dfrac{h}{\lambda}$. This relation links the particle (momentum) and wave (wavelength) pictures.

De Broglie proposed that every moving particle has a wavelength $\lambda=\dfrac{h}{mv}$. Heavier or faster particles have shorter wavelengths: at equal speeds an alpha particle has a shorter wavelength than a proton, neutron or electron. A freely falling body speeds up, so its wavelength keeps decreasing. An electron accelerated through $V$ volts has $\lambda\approx\dfrac{12.27}{\sqrt V}$ angstrom, of the order of X-ray wavelengths.

Key formulas

• $E=hf=hc/\lambda$
• $hf=\phi+K.E_{max}$ (Einstein's photoelectric equation)
• $K.E_{max}=eV_o$; $\phi=hf_o$
• $p=h/\lambda$; $\lambda_{de\,Broglie}=h/mv=h/\sqrt{2mK.E}$
• Stefan: $E\propto T^4$

Common MDCAT traps

• Units of $h$ are J s, not J s$^{-1}$; value $10^{-34}$, not $10^{34}$.
• Stopping potential depends on frequency, never on intensity.
• Convert grams to kilograms in $\lambda=h/mv$ (1 g = $10^{-3}$ kg).
• If photon energy just equals the work function, electrons are freed with zero K.E. and no current flows.
• Stopping potential in volts equals $K.E_{max}$ in eV, not the photon energy, unless the work function is zero.

Quick revision

• $h=6.63\times10^{-34}$ J s.
• Black body: continuous spectrum, both perfect emitter and absorber.
• Violet photons have the most energy among visible colours.
• Photoelectric effect is evidence for photons.
• Photon momentum $=h/\lambda$; rest mass zero.
• 1 eV $=1.6\times10^{-19}$ J.