Читать книгу Quantum Mechanical Foundations of Molecular Spectroscopy - Max Diem - Страница 14

In 1905, Einstein reported experimental results that further demonstrated the energy quantization of light. In the photoelectric experiment, light of variable color (frequency) illuminated a photocathode contained in an evacuated tube. An anode in the same tube was connected externally to the cathode through a current meter and a source of electric potential (such as a battery). Since the cathode and anode were separated by vacuum, no current was observed, unless light with a frequency above a threshold frequency was illuminating the photocathode. Einstein correctly concluded that light particles, or photons, with a frequency above this threshold value had sufficient kinetic energy to knock out electrons from the metal atoms of the photocathode. These “photoelectrons” left the metal surface with a kinetic energy given by

(1.9)

where ϕ is the work function, or the energy required to remove an electron from metal atoms. This energy basically is the atoms' ionization energy multiplied by Avogadro's number. Furthermore, Einstein reported that the photocurrent produced by the irradiation of the photocathode was proportional to the intensity of light, or the number of photons, but that increasing the intensity of light that had a frequency below the threshold did not produce any photocurrent. This provided further proof of Eq. (1.9).

This experiment further demonstrated that light has particle character with the kinetic energy of the photons given by Eq. (1.7), which led to the concept of wave–particle duality of light. Later, de Broglie theorized that the momentum p of a photon was given by

(1.10)

Equation (1.10) is known as the de Broglie equation. The wave–particle duality was later (1927) confirmed to be true for moving masses as well by the electron diffraction experiment of Davisson and Germer [3]. In this experiment, a beam of electrons was diffracted by an atomic lattice and produced a distinct interference pattern that suggested that the moving electrons exhibited wave properties. The particle–wave duality of both photons and moving matter can be summarized as follows.

For photons, the wave properties are manifested by diffraction experiments and summarized by Maxwell's equation. As for all wave propagation, the velocity of light, c, is related to wavelength λ and frequency ν by

(1.11)

with c = 2.998 × 10⁸ [m/s] and λ expressed in [m] and ν expressed in [Hz = s⁻¹]. The quantity is referred to as the wavenumber of radiation (in units of m⁻¹ or cm⁻¹) that indicates how many wave cycles occur per unit length:

(1.12)

The (kinetic) energy of a photon is given by

(1.13)

with ħ = h/2π and ω, the angular frequency, defined before as ω = 2πν.

From the classical definition of the momentum of matter and light, respectively,

(1.14)

it follows that the photon mass is given by

(1.15)

Notice that a photon can only move at the velocity of light and the photon mass can only be defined at the velocity c. Therefore, a photon has zero rest mass, m₀.

Particles of matter, on the other hand, have a nonzero rest mass, commonly referred to as their mass. This mass, however, is a function of velocity v and should be referred to as m_ν, which is given by

(1.16)

Example 1.1 Calculation of the mass of an electron moving at 99.0 % of the velocity of light (such velocities can easily be reached in a synchrotron).

Answer:

According to Eq. (1.16), the mass m_v of an electron at ν = 0.99 c is

(E1.1.1)

The electron at 99 % of the velocity of light has a mass of about seven times its rest mass.

Equation (1.16) demonstrates that the mass of any matter particle will reach infinity when accelerated to the velocity of light. Their kinetic energy at velocity ν (far from the velocity of light) is given by the classical expression

(1.17)

The discussion of the last paragraphs demonstrates that at the beginning of the twentieth century, experimental evidence was amassed that pointed to the necessity to redefine some aspects of classical physics. The next of these experiments that led to the formulation of quantum mechanics was the observation of “spectral lines” in the absorption and emission spectra of the hydrogen atom.