The observation of the photoelectric effect and the absorption/emission spectra of the hydrogen atom and the modifications required to formulate the blackbody emission theory were the triggers that forced the development of quantum mechanics. As pointed out in the introduction, the development of quantum mechanics is based on postulates, rather than axioms. The form of some of these postulates can be visualized from other principles, but their adoption as “the truth” came from the fact that they produced the correct results.
References
1 1 Halliday, D. and Resnick, R. (1960). Physics. New York: Wiley.
2 2 Engel, T. and Reid, P. (2010). Physical Chemistry, 2e. Upper Saddle River, NJ: Pearson Prentice Hall.
3 3 Davisson, C.J. and Germer, L.H. (1928). Reflection of electrons by a crystal of nickel. Proceedings of the National Academy of Sciences of the United States of America 14 (4): 317–322.
Problems
1 What is the maximum wavelength of electromagnetic radiation that can ionize an H atom in the n = 2 state?
2 Why is it that any photon with a wavelength below the limiting value obtained in (1) can ionize the H atom, whereas in standard spectroscopy, only a photon with the correct energy can cause a transition?
3 Assume that you carry out the experiment in (1) with light with a wavelength of 10 nm less than calculated in (1). What is the kinetic energy of the photoelectron created?
4 What is the velocity of the electron in Problem 3?
5 Using the de Broglie relation for matter waves, calculate the velocity to which an electron needs to be accelerated such that its wavelength is 10 nm.
6 What percentage of the velocity of light is the velocity in (5)?
7 What is the relativistic mass of this electron?
8 At what velocity is the wavelength of an electron 30 nm?
9 What is the momentum of such the electron in (8)?
10 What is the mass of a photon with a wavelength of 30 nm?
11 What is the momentum of the photon in (10)?
12 Compare and comment on the masses and momenta of the moving particles in Problems (8)–(11).
13 “Frequency doubling” or “second harmonic generation (SHG)” is a little optical trick (Appendix 3) in which two photons of the same wavelength are squashed into a new photon, while the energy is conserved. Calculate the wavelength of the photon created from frequency doubling of two photons with λ = 1064 nm.
14 “Sum frequency generation (SFG)” is another optical trick (Appendix 3) in which two photons of different wavelengths are squashed into a new photon while the energy is conserved. Calculate the wavelength of SFG photon created from combining two photons with λ1 = 1064 nm and λ2 = 783 nm.
15 The value of the Rydberg constant, Ry, can be calculated according to where e' = e/√4πεo and where mR is the reduced mass of electron and proton. Perform an analysis of the units of Ry.
16 Which two experiments demonstrate that light has wave and particle character?
17 Which two experiments demonstrate that moving electrons have wave and particle character?
2 Principles of Quantum Mechanics
Quantum mechanics presents an approach to describe the behavior of microscopic systems. Whereas in classical mechanics the position and momentum of a moving particle can be established simultaneously, Heisenberg's uncertainty principle prohibits the simultaneous determination of those two quantities. This is manifested by Eq. (2.1):
which implies that the uncertainty in the momentum and position always exceeds ħ/2, where ħ is Planck's constant divided by 2π. Mathematically, Eq. (2.1) follows from the fact that the operators responsible for defining position and momentum,
The incorporation of this uncertainty into the picture of the motion of microscopic particles leads to discrepancies between classical and quantum mechanics: classical physics has a deterministic outcome, which implies that if the position and velocity (trajectory) of a moving body are established, it is possible to predict with certainty where it is going to be found in the future. This principle certainly holds at the macroscopic scale: if the position and trajectory of a macroscopic body, for example, the moon, are known, it is certainly possible to calculate its position six days from now and to send a spaceship to this predicted position.
Quantum mechanical systems, on the other hand, obey a probabilistic behavior. Since the position and momentum can never be determined simultaneously at any point in time, the position (or momentum) in the future cannot be precisely predicted, only the probability of either of them. This is manifested in the postulate that all properties, present or future, of a particle are contained in a quantity known as the wavefunction Ψ of a system. This function, in general, depends on spatial coordinates and time; thus, for a one‐dimensional motion (to be discussed first), the wavefunction is written as Ψ(x, t). The probability of finding a quantum mechanical system at any time is given by the integral of the square of this wavefunction: ∫Ψ(x, t)2 dx. This is, in fact, one of the “postulates” on which quantum mechanics is based to be discussed next. Different authors list these postulates in different orders and include different postulates necessary for the description of quantum mechanical systems [1]. Quantum mechanics is unusual in that it is based on postulates, whereas science, in general, is axiom‐based.
2.1 Postulates of Quantum Mechanics
Postulate 1: The state of a quantum mechanical system is completely defined by a wavefunction Ψ(x, t). The square of this function, or in the case of complex wavefunction, the product Ψ*(x, t) Ψ(x, t), integrated over a volume element dτ (= dx dy dz in Cartesian coordinates or sin2θ dθ dφ in spherical polar coordinates) gives the probability of finding a system in the volume element dτ. Here, Ψ*(x, t) is the complex conjugate of the function Ψ(x, t). This postulate contains the transition from a deterministic to probabilistic description of a quantum mechanical system. The wavefunctions must be mathematically well behaved, that is, they must be single‐valued, continuous, having a continuous first derivative, and integratable (so they can be normalized).
Postulate