[...]... transition to modern quantum mechanics, and this early quantum theory came to be called the Old Quantum Theory 1.2 Wave–Particle Duality and the Uncertainty Relation The difficulty with the old quantum theory led many to search for some other type of theory, and some of the crucial steps along the way were due to de Broglie and Heisenberg who deepened the gulf between classical theory and modern theory... Theory C.11 Relativistic Quantum Mechanics and Particle Theory 319 319 319 321 322 323 324 326 329 330 330 330 D The Fundamental Physical Constants, 1986 303 303 306 306 333 Bibliography 335 Index 337 1 The Foundations of Quantum Physics 1.1 The Prelude to Quantum Mechanics Quantum mechanics is essentially a 20th century development... exponent, so we √ use 1/ 2π in each 20 QUANTUM MECHANICS Foundations and Applications and (∆x) = 2 , (1.14) (∆p) = 1.3.2 x2 − x p2 − p 2 (1.15) Operators It is possible for every ψ(x) that is continuous and has a continuous first derivative to calculate the corresponding φ(p) (although it may be difficult), and use ψ(x) to calculate the expected value of every function of x and use φ(p) to calculate the expected... these experiments have semiclassical explanations, the triumph of quantum mechanics is that it gives precise verification for an overwhelming number of experimental observations Its extensions into relativistic quantum mechanics through quantum electrodynamics (QED), electro-weak theory, and quantum chromodynamics (QCD) have led us to the Standard Model of today While the number of unanswered questions... positive to negative, but generally not both), and sends a pulse to the counter each time the trigger fires, 2 A digital counter that simply adds the number of pulses it receives, and 3 A timed gate that determines how long a period of time the counter will accept pulses This gate sets the count to zero when it starts, and stops 12 QUANTUM MECHANICS Foundations and Applications the counter from accepting any... determined by the diffraction pattern and we can assume the photon probably landed in the first diffraction ring This leaves us with an uncertainty of position, from physical optics, of ∆x ≈ fλ D where f is the focal length of the lens, D is the diameter of the lens, and λ is the wavelength of the photon In the reflection of the photon from the 14 QUANTUM MECHANICS Foundations and Applications Incident light ... ), such that E/A = EBB where EBB (λ, T ) is the emissivity of a black body at that wavelength and temperature 2 The classical expression relating energy density and momentum for plane waves relate the pressure P and the energy density U for isotropic radiation: 1 P = 3U 4 QUANTUM MECHANICS Foundations and Applications 3 From classical electromagnetic theory, it then follows that the total power, eB... Planck assumed that the possible radiating energies were given by En = nhν 6 QUANTUM MECHANICS Foundations and Applications where ν(= f ) is the frequency of the oscillator, n is an integer, and h is a constant Thus he arrived at the expression I(λ, T ) = 2πc2 h λ5 (ech/λKT − 1) (1.3) The constant h is called Planck’s constant, and has the value h = 6.626×10−34 J-s Problem 1.2 Show from Equation (1.2)... 2 ν= (1.9) h 8 0 h (1 + m/M ) n 2 n and the wave number is given by ν= ˜ ν 1 = = Z 2R c λ 1 1 − 2 n n2 , (1.10) where R is the Rydberg constant, given by R= me4 8 2 ch3 (1 + m/M ) 0 (1.11) 10 QUANTUM MECHANICS Foundations and Applications Its value for hydrogen is RH = 10973731.5 m−1 An empirical formula of this kind was discovered in 1885 by Balmer for n = 2, and all of the spectral lines for n... 2m (1.1) 1 2 QUANTUM MECHANICS Foundations and Applications where B is the magnetic induction, e is the electronic charge, and m is the electron mass While many atoms exhibited this behavior, notably the sodium D-lines that were studied by Zeeman, numerous examples were discovered that did not follow any simple pattern Some had many additional frequencies, some had no unshifted frequency, and sometimes .