From Classical Mechanics to Quantum Mechanics
7.4 Quantum Hypotheses
140 7 From Classical Mechanics to Quantum Mechanics
0.0 5.0×1014 1.0×1015 1.5×1015 2.0×1015 ν (s-1)
0.0 5.0×10-16 1.0×10-15 1.5×10-15 2.0×10-15
u (J s m-3 )
T = 3,800 K T = 4,800 K T = 5,800 K
Range of visible light
Fig. 7.10 Spectral energy density of the black body at different temperatures. The valueT=5800 K corresponds to the surface temperature of the sun
Compton Effect
When X-rays of a given frequency interact with atoms and are scattered with an angleψ with respect to the direction of incidence, the frequency of the emerging rays is found to depend onψ. This outcome is in contrast with the prediction of the electromagnetic theory, according to which the frequency of the scattered rays should be equal to that of the impinging ones. The dependence of frequency on the scattering angle is also calledCompton effect.
The experimental setup for measuring the Compton effect is schematically shown in Fig.7.11. The gray box in the middle of the figure is a piece of solid material, onto which the radiation impinges from the left (dark arrows); the vertical lines are the intersections of the constant-phase planes with the plane of the figure. The gray arrows on the right represent the part of the radiation that does not interact with the material and exits from it unaltered, while the white arrows indicate some of the directions of the rays scattered by the material. The circumferences are the intersections with the figure’s plane of the spherical waves produced by the scattering. The origin of the measuring angle is aligned with the direction of the incoming radiation, so that ψ=0 corresponds to the absence of scattering,ψ=2πto reflection.
Fig. 7.11 Scheme of the experimental setup for measuring the Compton effect
ψ
In essence, the contradictions with the physical laws known in the early 1900s were found thanks to the refinement of the experimental techniques. Such refinements were in turn made available by the general advancement of science that had taken place in the preceding decades. Thanks to them, it was possible to start investigating the microscopic world, namely, the dynamics of elementary particles. A parallel improvement took place in the same years in the investigation of the dynamics at high velocities, and led to the concepts of Special Relativity (1905).13
7.4.1 Planck’s Solution of the Black-Body Problem
To explain the features of the black-body radiation, Planck made in 1900 the hy- pothesis that a monochromatic electromagnetic energy is absorbed or emitted only in quantities that are integer multiples of a fixed quantityhν, wherehis a suitable constant [82]. The occupation number then becomes
Pn=P0 exp (−nβ h ν), β =1/(kBT). (7.17) As a consequence, using the same procedure as in Sect. 6.6.2 after replacing the integrals with sums, yields for the average energyn h νthe expression14
n h ν= ∞
n=0n h ν Pn ∞
n=0Pn = h ν
exp (β h ν)−1. (7.18)
In contrast with the constant valuekBT used in the determination of the Rayleigh- Jeans law, here the average energy of each monochromatic component depends on the
13As the particles’ velocities that occur in solid-state physics are low, Special Relativity is not used in this book; the only exception is in the explanation of the Compton effect, illustrated in Sect.
7.4.3.
14The detailed calculation leading to (7.18) is shown in Prob. 6.1.
142 7 From Classical Mechanics to Quantum Mechanics component’s frequency. Multiplying (7.18) by the number 8π ν2/c3of monochro- matic components of the electromagnetic field per unit volume and frequency, found in Sect.7.3, yields for the spectral energy density of the black body the expression
u(ν,T)=8π h ν3/c3
exp [h ν/(kBT)]−1, (7.19) calledPlanck law(1900). The derivation of (7.19) involves one undetermined param- eter,h. If the latter is made equal to the Planck constant introduced in the description of the photoelectric effect (Sect.7.3), the resulting expression fits perfectly the exper- imental data like those of Fig.7.10. Remembering that the spectral energy density of a black body in equilibrium is a universal function, it follows thathdoes not depend on the specific experiment, namely, it is a universal constant.
The low-frequency limit of (7.19),h ν≪kBT, is independent ofhand renders the Rayleigh–Jeans law (7.16).
7.4.2 Einstein’s Solution of the Photoelectric Effect
In 1905, Einstein proposed the following explanation of the photoelectric effect:
the transport of electromagnetic energy isquantized; specifically, a monochromatic electromagnetic wave of frequencyνis made of the flux of identical objects, called photons, each carrying the energyh ν. In the interaction with a photon, an electron may absorb an energy up toh ν. If the absorbed energy is exactlyhν, the photon is annihilated [34].15This theory provides a correct explanation of the photoelectric effect: with reference to Fig.7.7, the photoelectric current increases as the spectral power increases at constantν, because the number of photons is larger: as a consequence, the number of photoelectrons is larger as well. In turn, with reference to Fig.7.8, the blocking voltageVRincreases asνincreases at constant, because the photons are more energetic; however, they are fewer, which explains why the curves intersect each other: the spectral power, in fact, can be written as=dE/(dtdν)= h ν[dN/(dtdν)], where the quantity in brackets is the number of photons per unit time and frequency; as a consequence, the constraint=const of the experiment of Fig.7.8makes the quantity in brackets to decrease when the photon energyh ν increases.
7.4.3 Explanation of the Compton Effect
The concept of photon, introduced in Sect.7.4.2, explains the Compton effect by describing the interaction of the electron with the electromagnetic field as the collision
15Einstein’s hypothesis is more general than Planck’s: the latter, in fact, assumes that energy is quantized only in the absorption or emission events.
between the electron and a photon [19]. As the photon’s velocity isc, its rest mass is zero (Sect. 3.13.7); in turn, the modulus of the photon’s momentum isp=E/c, which is consistent with classical electromagnetism (compare with (5.43)).
The analysis of the electron-phonon collision is worked out assuming that the system made of the two particles under consideration is isolated; thus, the calculation is based upon the energy- and momentum-conservation equations, and the results of Sect. 3.13.8 hold. The dynamical quantities for the photon are given by
E=h ν, p= E c = h ν
c = h
λ, (7.20)
the second of which derives from (5.55) expressedin vacuo. Defining thereduced Planck constanth¯ =h/(2π)≃1.055×10−34 J s, and using the moduluskof the wave vector, (7.20) becomes
E= ¯h2π ν= ¯h ω, p= h¯
λ/(2π) = ¯h k. (7.21) The second relation of (7.21) in vector form reads
p= ¯hk. (7.22)
Here the useful outcome of the analysis of Sect. 3.13.8 is (3.92), that relates the photon’s energies prior and after the collision (Ea andEb, respectively) with the deflection angleψ(Fig. 3.7). UsingE=c h/λin (3.92) yields
λb−λa =2λ0 sin2 ψ
2
, λ0= h
m0c, (7.23)
withλ0 ≃2.43×10−12 m theCompton wavelength(1923). The frequency corre- sponding to it isν0=c/λ0 ≃1.2×1020Hz. The maximum difference in wavelength corresponds to the case of reflection, max(λb−λa)= 2λ0. Even in this case, the smallness ofλ0makes the effect difficult to measure; in practice, the shift in wave- length is detectable only for sufficiently small values ofλa, typically in the range of 10−10 m corresponding to the X-ray frequencies (ν ∼1018 s−1). Due to the large energy of the photon, the energy transferred to the electron brings the latter into a high-velocity regime; this, in turn, imposes the use of the relativistic expressions for describing the electron’s dynamics.
7.4.4 Bohr’s Hypothesis
The description of the monochromatic components of the electromagnetic field as a flow of identical photons with energyh νlends itself to the explanation of the Balmer law (7.4). Such an explanation (Bohr’s hypothesis, 1913) is based on the idea that, if
144 7 From Classical Mechanics to Quantum Mechanics νnmis the frequency of the emitted radiation, the corresponding energy of the emitted photon ish νnm; multiplying (7.4) byhand remembering thatm > nthen yields
h νnm=h νR
1 n2 − 1
m2
=
−h νR
m2
−
−hνR
n2
. (7.24)
As the left hand side is the energy of the emitted photon, the terms on right hand side can be recast as
Em= −hνR
m2 , En= −hνR
n2 , En< Em<0 ; (7.25) then, ifEm(En) is interpreted as the atom’s energy before (after) emitting the photon, Balmer’s law becomes the expression of energy conservation. From this, the emission rule of Ritz is easily explained; in fact, (7.5) is equivalent to
Em−En=(Em−Ek)+(Ek−En). (7.26) Bohr’s hypothesis is expressed more precisely by the following statements:
1. The energy variations of the atom are due to the electrons of the outer shell, that exchange energy with the electromagnetic field.
2. The total energy of a non-radiative state is quantized, namely, it is associated to an integer index: En = −h νR/n2,n = 1, 2,. . .; the values of energy thus identified are calledenergy levels. The lowest level corresponds ton=1 and is calledground levelorground state.
3. The total energy can vary only between the quantized levels by exchanging with the electromagnetic field a photon of energyνnm=(Em−En)/ h.
It is interesting to note that, by combining Bohr’s hypothesis with the planetary model of the atom, the quantization of the other dynamical quantities follows from that of energy; again, the case of a circular orbit is considered. By way of example, usingEn = −h νR/n2 in the second relation of (7.3) provides the quantization of the orbit’s radius:
r =rn= − q2
8π ε0En = q2 8π ε0
n2
h νR. (7.27)
The smallest radiusr1 corresponds to the ground staten=1; takingνRfrom (7.4) and the other constants from Table D.1 one findsr1≃0.05 nm; despite the simplicity of the model,r1is fairly close to the experimental valueragiven in (7.1).
In turn, the velocity is quantized by combining (7.3) to obtainT = −V /2= −E;
replacing the expressions ofT andEthen yields 1
2mu2= h νR
n2 , u=un=
2h νR
m n2 . (7.28)
The largest velocity is found from (7.28) by lettingn=1 and using the minimum value for the mass, that is, the rest massm = m0. It turns outu1 ≃ 7×10−3c;
as a consequence, the velocity of a bound electron belonging to the outer shell of the atom can be considered non relativistic. Thanks to this result, from now on the electron’s mass will be identified with the rest mass. Finally, for the angular momentumM=r p=r muone finds
M=Mn= q2n2 8π ε0h νR m
2h νR
m n2 = 1 2π
q2 ε0
m 8h νR
n. (7.29)
The quantity in brackets in (7.29) has the same units asM, namely, an action (Sect.
1.5) and, replacing the constants, it turns out16 to be equal toh. Using the reduced Planck constant it follows
Mn=nh.¯ (7.30)
The Bohr hypothesis provides a coherent description of some atomic properties; yet it does not explain, for instance, the fact that the electron belonging to an orbit of energyEn= −h νR/n2does not radiate, in contrast to what is predicted by the elec- tromagnetic theory (compare with the discussion in Sect.7.3). Another phenomenon not explained by the hypothesis is the fact that only the ground state of the atom is stable, whereas the excited states are unstable and tend to decay to the ground state.
7.4.5 De Broglie’s Hypothesis
The explanation of the Compton effect (Sect.7.4.3) involves a description of the photon’s dynamics in which the latter is treated like a particle having energy and momentum. Such mechanical properties are obtained from the wave properties of a monochromatic component of the electromagnetic field: the relations involved are (7.20) (or (7.21)), by which the photon energy is related to the frequency, and its momentum to the wave vector. It is worth specifying that such relations are applied to the asymptotic part of the motion, namely, when the photon behaves like a free particle. In 1924, de Broglie postulated that analogous relations should hold for the free motion of a real particle: in this case, the fundamental dynamic properties are energy and momentum, to which a frequency and a wavelength (or a wave vector) are associated by relations identical to (7.20), (7.21),17
ω=2π ν =2π E h =E
¯
h, k= 2π λ = 2π
h/p =p
¯
h, k=p
¯
h. (7.31) The usefulness of associating, e.g., a wavelength to a particle’s motion lies in the possibility of qualitatively justifying the quantization of the mechanical properties
16This result shows that the physical constants appearing in (7.29) are not independent from each other. Among them,νRis considered the dependent one, whileq,m=m0,ε0, andhare considered fundamental.
17The wavelength associated to the particle’s momentum is calledde Broglie’s wavelength.
146 7 From Classical Mechanics to Quantum Mechanics illustrated in Sect. 7.4.4. For this, consider the case of the circular orbit of the planetary motion, and associate a wavelength to the particle’s momentum,λ=h/p.
Such an association violates the prescription that (7.31) apply only to a free motion;
however, if the orbit’s radius is very large, such that λ ≪ r, the orbit may be considered as locally linear and the concept of wavelength is applicable. Replacing λ=h/pin (7.30) yields
2π r=n λ, (7.32)
namely, the quantization of the mechanical properties implies that the orbit’s length is an integer multiple of the wavelength associated to the particle. This outcome suggests that the formal description of quantization should be sought in the field of eigenvalue equations.
De Broglie also postulated that a functionψ=ψ(r,t), containing the parameters ω,kdefined in (7.31), and calledwave function, is associated to the particle’s motion.
Its meaning is provisionally left indefinite; as for its form, it is sensible to associate to the free motion, which is the simplest one, the simplest wave function, that is, the planar monochromatic wave. The latter is conveniently expressed in complex form as
ψ=Aexp [i (k·r−ω t)], (7.33)
whereA=0 is a complex constant, not specified. Due to (7.31), the constant wave vectorkidentifies the momentum of the particle, and the angular frequencyωiden- tifies its total energy, which in a free motion coincides with the kinetic energy. It is worth pointing out that, despite its form, the wave function is not of electromagnetic nature; in fact, remembering that in a free motion it isH=p2/(2m)=E, withH the Hamiltonian function, it follows
¯
h ω= 1
2mh¯2k2, ω(k)= ¯h
2mk2, (7.34)
which is different from the electromagnetic relationω =c k. By the same token it would not be correct to identify the particle’s velocity with the phase velocityuf
derived from the electromagnetic definition; in fact, one has uf =ω
k = E/h¯
p/h¯ =p2/(2m)
p = p
2m. (7.35)
The proper definition of velocity is that deriving from Hamilton’s Eqs. (1.42); its ith component reads in this case
ui= ˙xi= ∂H
∂pi = 1
¯ h
∂H
∂ki = ∂ω
∂ki = ¯hki m =pi
m. (7.36)
The concepts introduced so far must now be extended to motions of a more general type. A sensible generalization is that of the conservative motion of a particle subjected to the force deriving from a potential energy V(r). In this case the association described by (7.31) works only partially, because in a conservative
motion the total energy is a constant, whereas momentum is generally not so. As a consequence, lettingω=E/h¯yields for the wave function the form
ψ=w(r) exp (−iω t), (7.37) which is still monochromatic but, in general, not planar. Itsspatial part w(r) reduces toAexp (ik·r) for the free motion. The function of the form (7.37) is postulated to be the wave function associated with the motion of a particle at constant energy E= ¯hω. While the time dependence ofψis prescribed, its space dependence must be worked out, likely by solving a suitable equation involving the potential energy V, the particle’s mass and, possibly, other parameters.