Download Quantum Mechanics I SS 2017 - Institut für Theoretische Physik

Quantum Mechanics I
SS 2017
Roser Valentı́
Institut für Theoretische Physik
This script has been prepared by Roser Valentı́ on the basis of the following literature:
1) W. Nolting: ”Grundkurs Theoretische Physik 5/1. Quantenmechanik-Grundlagen”,
Springer 2004.
2) F. Schwabl: ”Quantenmechanik”, Springer 1993.
3) David J. Griffiths: ”Introduction to Quantum Mechanics”, Pearson education 2005.
4) S. Gasiorowicz: ”QuantenPhysik”, Oldenburg 2005.
5) J. J. Sakurai: ” Modern Quantum Mechanics”, Addison-Wesley Publishing Company 1994.
6) R. Shankar: Principles of Quantum Mechanics, Second Edition Springer 2008.
7) C. Cohen-Tannoudji, B. Diu, F. Laloke: ”Quantum Mechanics ”, Wiley-VCH 2005.
I would like to thank Kateryna Foyevtsova for typing most of the latex version of the
manuscript.
Contents
1 Basics in quantum mechanics
1.1 Historical introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Particle nature of electromagnetic waves . . . . . . . . . . . . . . . .
1.2.1 Blackbody radiation / cavity radiation . . . . . . . . . . . . .
1.2.2 Photoelectric effect (Hertz 1887, Hallwachs 1888, Lenard 1900)
1.2.3 Compton - Effect (Compton 1924) . . . . . . . . . . . . . . .
1.3 Wave nature of particles . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Two-slit experiment: wave-particle dualism . . . . . . . . . . .
1.4 Discrete states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Quantization of spin/angular momentum . . . . . . . . . . . .
1.4.2 Energy quantization . . . . . . . . . . . . . . . . . . . . . . .
2 Wavefunctions, operators and the Schrödinger equation
2.1 Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Statistical interpretation of the wavefunction . . .
2.1.2 Planewave . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Wavepacket . . . . . . . . . . . . . . . . . . . . . .
2.2 The Schrödinger equation . . . . . . . . . . . . . . . . . .
2.2.1 Hamilton operator . . . . . . . . . . . . . . . . . .
2.2.2 Correspondence principle . . . . . . . . . . . . . . .
2.2.3 Properties of operators and wavefunctions . . . . .
2.2.4 Hermitian operator . . . . . . . . . . . . . . . . . .
2.3 Norm conservation and the continuity equation . . . . . .
2.4 Expectation values of operators . . . . . . . . . . . . . . .
2.5 Coordinate and momentum representations . . . . . . . . .
2.6 Ehrenfest theorem . . . . . . . . . . . . . . . . . . . . . .
2.7 Heisenberg’s uncertainty principle . . . . . . . . . . . . . .
2.8 Time independent Schrödinger equation . . . . . . . . . .
3 Representation theory
3.1 State vectors and the Dirac representation
3.1.1 Hilbert space. Scalar product . . .
3.1.2 Operators in the Hilbert space . . .
3.2 Representation of an operator . . . . . . .
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2
CONTENTS
3.2.1
3.3
3.4
Change of basis for operators and vectors
formation . . . . . . . . . . . . . . . . .
3.2.2 Spectral representation of an operator .
3.2.3 Complete set of conmuting observables .
Time evolution . . . . . . . . . . . . . . . . . .
3.3.1 Schrödinger picture . . . . . . . . . . . .
3.3.2 Heisenberg picture . . . . . . . . . . . .
3.3.3 Dirac (interaction) picture . . . . . . . .
3.3.4 Example: two level system . . . . . . . .
Postulates in quantum mechanics.
Density matrix . . . . . . . . . . . . . . . . . .
3.4.1 Postulates in Quantum Mechanics . . . .
3.4.2 Density matrix . . . . . . . . . . . . . .
3.4.3 Properties of the density matrix . . . . .
through
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4 One dimensional examples in Quantum Mechanics
4.1 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . .
4.1.1 Ladder operators . . . . . . . . . . . . . . . . . . . . .
4.1.2 Eigenfunctions of Ĥ (Hermite polynomials) . . . . . . .
4.1.3 Recurrence relations . . . . . . . . . . . . . . . . . . .
4.1.4 Parity operator . . . . . . . . . . . . . . . . . . . . . .
4.1.5 Time dependent Schrödinger equation. Coherent states
4.2 Free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Periodic boundary conditions . . . . . . . . . . . . . .
4.2.2 Square well of infinite depth . . . . . . . . . . . . . . .
4.2.3 bound states ⇔ scattering states . . . . . . . .
4.2.4 Potential step . . . . . . . . . . . . . . . . . . . . . . .
4.3 Tunnel effect . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Scattering matrix . . . . . . . . . . . . . . . . . . . . .
4.4 Some examples of tunneling . . . . . . . . . . . . . . . . . . .
4.4.1 Alpha-radioactivity . . . . . . . . . . . . . . . . . . . .
4.4.2 Metal . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Kronig-Penney-Model . . . . . . . . . . . . . . . . . .
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5 Theory of angular momentum
5.1 Importance of angular momentum . . . . . . . . . . . . . .
5.1.1 Orbital angular momentum in Quantum Mechanics
5.1.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Generalization: angular momentum J~ . . . . . . . . . . . .
5.2.1 Special properties of J = S = 1/2 . . . . . . . . . .
5.2.2 Non-relativistic description of a spin-1/2 particle . .
5.3 Addition of angular momenta . . . . . . . . . . . . . . . .
5.3.1 Addition of two spins 1/2 . . . . . . . . . . . . . .
5.4 Symmetries, conservation laws, degeneracies . . . . . . . .
5.4.1 Symmetries in classical physics . . . . . . . . . . .
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CONTENTS
3
5.4.2
Symmetries in quantum mechanics . . . . . . . . . . . . . . . . . . 112
6 Gauge transformations and the Aharanov-Bohm effect
113
6.1 Gauge Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Aharanov-Bohm effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 Particles in a central potential
7.1 Coulomb potential . . . . . . . . . . . . . . . . . .
7.2 Angular momentum operator . . . . . . . . . . . .
2
~ˆ . . . . . . . . . . . . . . . . . .
7.2.1 Operator L
2
~ˆ
7.2.2 Eigenvalues and eigenfunctions of L̂z and L
2
~ˆ ] = 0 . . . . . . . . . . . . . . . . . .
7.2.3 [Ĥ, L
7.3 Radial equation . . . . . . . . . . . . . . . . . . . .
7.4 Bound states of the hydrogen atom . . . . . . . . .
7.5 Principal quantum number . . . . . . . . . . . . . .
7.5.1 Degeneracy . . . . . . . . . . . . . . . . . .
7.6 Eigenfunctions of the Hydrogen atom . . . . . . . .
7.7 The spectrum of Hydrogen . . . . . . . . . . . . . .
8 Identical particles
8.1 Two-particle systems . . . . . .
8.2 Bosons and fermions . . . . . .
8.3 Spin . . . . . . . . . . . . . . .
8.4 Pauli principle . . . . . . . . . .
8.5 Spatial function / Spin function
8.6 Helium (two identical electrons)
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9 Approximate methods
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9.1 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.2 Time-independent Perturbation theory . . . . . . . . . . . . . . . . . . . . 137
4
CONTENTS
Chapter 1
Basics in quantum mechanics
1.1
Historical introduction
End of 19th century.- Physics seemed to be understood:
• Newton’s mechanics could explain the motion on earth and in the universe.
• Hydrodynamics was well explained by Euler, Navier and Stokes.
• Maxwell equations provided the unification of electricity, magnetism and optics.
Light was described as an electromagnetic wave.
• Boltzmann settled the relation between thermodynamics and statistical mechanics.
Around this time (1874) Planck wanted to study physics and asked a professor at the University of Munich (von Jolly) which were the prospects of research in physics. Von Jolly
told him that physics was a complete science with little prospects of further development.
This statement didn’t impede Planck from studying physics. He became the founder of
Quantum Mechanics and in 1918 he received the Nobel Prize in physics.
In fact in the late 19th / beginning 20th century, there were a few experimental observations
that could not be explained with the existing classical theories. We will discuss some of
these observations in this chapter:
1. Blackbody radiation (Planck)
2. Photoelectric effect (Einstein)
3. Compton effect (Compton)
4. Discrete spectra (Bohr)
We shall start with a discussion that had gone over centuries, namely which is the nature
of light:
→ Newton (1643 - 1727): Light consists of particles
5
6
CHAPTER 1. BASICS IN QUANTUM MECHANICS
→ Huygens (1629 - 1695), Fresnel (1788 - 1827): Light is a wave
→ Early 20th century: Light consists of particles (Photoelectric effect, Compton effect)
The origin of this controversy is that in classical physics there are two well defined entities:
particles and waves.
Particles are localized objects of energy and momentum and they are described at any
time, t by the position q(t) and momentum p(t).
Waves are disturbances spread over space. A wave is described by a wavefunction Ψ(r, t)
which characterizes the disturbance at the point r and time t.
The solution of the controversy on the nature of light is given by a concept that will be
introduced in this course which is inherent in Quantum Mechanics: dualism wave-particle.
1.2
1.2.1
Particle nature of electromagnetic waves
Blackbody radiation / cavity radiation
A black body is an idealized system that acts as a perfect absorber and a perfect radiator
of electromagnetic waves. A black body at temperature T absorbs all wavelengths λ of
energy and radiates (emits) at all wavelengths. λ = 2πc/ω with c light velocity and ω
angular frequency. A good example of a black body is a furnace.
Let’s assume we have a cavity with black walls. After a short period of time the radiation
inside the cavity reaches thermal equilibrium caused by the emission and absorption by
the walls. When this equilibrium is reached, the radiation field no longer varies and the
electromagnetic radiation is described by standing waves.
We want to find the energy of radiation per unit volume in the cavity as a function
of angular frequency ω = 2πν and temperature T , u(ω, T ) = Energy/V olume. By
combining classical electrodynamics and classical statistical mechanics one obtains that
the density of radiation energy as a function of T and ω follows:
kB T
← rayleigh-jeans law
u(ω, T ) = 2 3 ω 2
π c
This law predicts that the total energy per unit volume over the whole range of frequencies
diverges, what is against the experimental observations.
R
u(ω)d ω = ∞ diverges! This is called ”ultraviolet catastrophy” since it means a
continuous increase in radiated energy with frequency.
1.2. PARTICLE NATURE OF ELECTROMAGNETIC WAVES
7
Experimentally, one observes that u(ω) has the following dependence on ω:
at low ω the Rayleigh-Jeans Law is correct, but at high frequencies the observed u(ω)
deviates considerably from the Rayleigh-Jeans Law. It follows the Wien’s radiation Law:
lim u(ω, T ) = A ω 3 exp
ω→∞
−gω
T
(1.1)
with A and g constants.
Planck got interested in this problem and in 1900 he derived the expression:
u(ω, T ) =
kB T ω 2 h̄ω/kB T
π 2 c3 eh̄ω/kB T − 1
(1.2)
where kB is the Boltzmann constant.
He assumed that the energy exchange of the radiation with the walls does not take place
continuously but in discrete units of h̄ω:
εn = nh̄ω = nhν ,
↑
n∈N
quantisation of the radiation energy
The quantity h̄ is the Planck action constant,
h̄ = h/2π = 1.05 × 10−27 erg · s = 1.05 × 10−27 J · s
(1.3)
This is a very small quantity. This fact explains why the microscopically necessary energy
quantisation can be neglected for macroscopic phenomena, where classical physics works
fine.
The limits of the Planck radiation formula are:
1. for h̄ω/kB T ≪ 1, h̄ becomes negligible, the result approaches the classical law:
u(ω, T ) =
1
kB T ω
π 2 c3
2
← rayleigh-jeans law
This is an example of the correspondence principle.
2. for h̄ω/kB T ≫ 1,
u(ω, T ) =
h̄ω 3 −h̄ω/kB T
e
π 2 c3
← wien’s radiation formula
8
CHAPTER 1. BASICS IN QUANTUM MECHANICS
1.2.2
Photoelectric effect (Hertz 1887, Hallwachs 1888, Lenard
1900)
Consider a polished, negatively charged zinc plate exposed to ultraviolet light as in the
picture. What one observes is that the zinc plate looses its charge. The photoelectric
effect is the ejection of electrons from matter by incident electromagnetic radiation.
ν = ω/(2π) is the light radiation frequency and ve denotes the velocity of electrons.
One observes that:
1. The photoelectric effect happens at frequencies ν > νg , where νg is a threshold
frequency and νg is material dependent.
2. The kinetic energy of the emitted electron is determined by the frequency ν of the
incident light and independent of the intensity of light.
3. For ν ≥ νg , the number of emitted electrons is proportional to the intensity of
incident light.
4. The photoeffect shows no time retardation (< 10−9 s >).
Classically, the energy density of the electromagnetic field is:
ωEM =
1 2 ǫ0 2
B + E
2µ0
2
(1.4)
This equation shows that (i) there is no threshold frequency for the emission of electrons;
and (ii) a certain amount of time is necessary until enough energy is in the solid to push
out the electrons. Both (i) and (ii) contradict the experimental observations.
The solution to this discrepancy was given by Albert Einstein.
He postulated (Ann. Phys. 17, 132 (1905)) the light quanta hypothesis:
⋆
The electromagnetic radiation with frequency ν behaves, when interacting with
matter, as a collection of light quanta (photons) with the energy:
E = hν = h̄ω
with h: Planck’s constant and ν the radiation frequency.
1.2. PARTICLE NATURE OF ELECTROMAGNETIC WAVES
9
Each released electron from the solid adsorbs a light quanta of energy hν. This light
quanta is used to
(i) free the electron from the binding energy in the solid (WA ), also called work function
and
(ii) provide the electron with kinetic energy Ee = 21 mv 2 . Then:
1
h̄ω = mv 2 + WA .
2
(1.5)
The work function WA is material dependent and is of the order of a few electronvolts.
For a relativistic particle, the energy is:
E=
q
p2 c2 + m2 c4 .
(1.6)
Since the velocity of light is v = c then the rest mass of a photon is m = 0 and E = pc
Using Einstein’s hypothesis:
pc = E = h̄ω = h̄ck ⇒
p~ = h̄~k.
(1.7)
⋆
Light consists of photons with energy E = h̄ω, velocity c
and momentum p~ = h̄~k
This result goes beyond Planck’s postulate since it not only contains the quantized nature
of the radiation energy but also implies the particle nature of light.
Monochromatic light of energy E and frequency ω contains n = E/h̄ω photons.
1.2.3
Compton - Effect (Compton 1924)
This experiment is one of the most convincing observations about the particle-wave nature
of light. Let’s consider the elastic scattering of X-ray light with wavector ~k = (2π/λ) k̂
with an electron initially at rest.
10
CHAPTER 1. BASICS IN QUANTUM MECHANICS
Energy and momentum of the photon γ is given by:
Ephoton = h̄ω = h̄ck
p~photon = h̄~k.
Since the electron is at rest, through the scattering with the photon γ acquires a momentum p~.
Energy and momentum are conserved in this elastic process. The electron has to be
treated relativistically since the momenta changes are of the order of me c.
photon
electron
initial state
h̄~k + 0
h̄ck + me c2
photon
electron
final state
=
h̄k~′ + p~′
q
= h̄ck ′ + me 2 c4 + c2 p′ 2
h
h̄c(k − k ′ ) + me c2
i2
=
momentum conservation
energy conservation
q
me 2 c4 + c2 p′ 2
2
2
,
(1.8)
2
(1.9)
(p~′ )2 = h̄2 (~k − k~′ )2 = h̄2 (k 2 + k ′ − 2~k k~′ ).
(1.10)
2
2
h̄2 (k 2 − 2kk ′ + k ′ ) + 2h̄(k − k ′ )me c = h̄2 (k 2 + k ′ − 2~k k~′ ),
(1.11)
−2h̄2 kk ′ + 2h̄(k − k ′ )me c = −2h̄2~k k~′ .
(1.12)
~k k~′ = kk ′ cos θ.
(1.13)
h̄2 (k 2 − 2kk ′ + k ′ ) + 2h̄(k − k ′ )me c = p′
we substitute in the previous equation p′2 by:
2
Then:
Since:
Then:
k − k′ =
h̄
kk ′ (1 − cos θ)
me c
(1.14)
1.3. WAVE NATURE OF PARTICLES
11
In terms of the wavelength λ = 2π/k, we have that:
1
1
k − k = 2π
− ′
λ λ
and equation (1.12) can be written as:
′
λ′ − λ =
= 2π
λ′ − λ
,
λλ′
(1.15)
4πh̄
sin 2 θ/2 = 2λc sin 2 θ/2
me c
where λc = mhe c = 2.4263 × 10−2 A is the Compton wavelength.
λc is defined by three fundamental constants and has the dimension of length.
This experiment evidences the particle character of the electromagnetic radiation. In the
experiment one observes two maxima at λ and at λ′ with λ′ > λ.
In the classical limit (h̄ → 0) λ = λ′ .
Following classical physics, the electron would first get momentum and energy from the
electromagnetic field and accelerate. The energy would then be emitted again from the
moving electron at the same frequency, what is in contradiction with experiment.
→
→
Both experiments, Compton - and Photoelectric Effect, show the particle nature
of light.
On the other hand, light has also wave character, for instance in diffraction
experiments.
⋆ The quantum nature of light is relativistic → quantum electrodynamics. In this course
we will mainly restrict ourselves to non-relativistic quantum mechanics.
1.3
1.3.1
Wave nature of particles
Two-slit experiment: wave-particle dualism
Up to now we concentrated on the particle nature of electromagnetic radiation. Let’s now
consider the complementary case, i.e., a set of particles, for instance, electrons, for which
we investigate their wave nature.
12
CHAPTER 1. BASICS IN QUANTUM MECHANICS
We shall perform the so-called two-slit or double-slit experiment. Let’s first assume that
we have a light source and light diffracts through both slits. As is well-known, in the
detector we will observe a wave-like interference pattern. This interference pattern results in bright and dark regions, which correspond to the constructive and destructive
interference, respectively.
We will now consider this experiment with a source of electrons instead of light. We first
leave slit 1 open and slit 2 close as in the picture. A certain amount of electrons will
travel through slit 1 towards the detector.
When an electron hits the detector, we hear a ”click”, and this shows that electrons don’t
behave like waves but like particles and we expect a distribution of detected particles as
shown by P1 . We now close slit 1 and open slit 2, and a similar behaviour happens, with
P2 being the distribution of the detected particles, where the maximum of P2 is at the
same position level as the position of slit 2.
We now open the two slits at the same time. For classical particles, we expect
that the total distribution of particles will be the sum of the distributions P1 + P2 .
BUT, for an enough amount of electrons coming out of the source, P1 + P2 is not what
1.3. WAVE NATURE OF PARTICLES
13
one observes. What is observed is an interference pattern, like in optics:
This interference pattern can be only explained if we consider the electrons as waves, since
q
q
if we consider L+ = L2 + (d/2 + x)2 and L− = L2 + (x − d/2)2 to be the pathes that
the wave follows to reach the detector at x, then the condition for a maximum in the
interference pattern is:
L+ − L− = nλ
λ → “wavelength′′ of the electrons
thus
xmax ≈ nλL/d.
This experiment is a manifestation of the wave character of electrons. The experiment
can be done also with atoms, molecules,... i.e. any particle at the atomic level.
⋆ de Broglie:
Associates to each particle a wave with wavevector:
~k = p~
and frequency ω = E/h̄
h̄√
λ = h/ | p~ |
| p~ |= 2mE
From other experiments, like the photoelectric effect, we know that electrons have particle
14
CHAPTER 1. BASICS IN QUANTUM MECHANICS
nature.
Let’s consider the situation of performing a more precise observation. In fact, we want
to know through which slit the electron travels. In case that the electrons behave like
particles (small ”balls”), they will go either through slit 1 or slit 2. We can try to observe
that by shining light on the particles as shown in the picture:
If a particle travels through slit 1 (2), it will scatter the light upwards (downwards). The
observation of the scattered light in the upper or lower part of the detector plus the
knowledge of the “click” when the particle hits the detector will be enough to tell through
which slit the particle went. When these two observations are fulfilled, the interference
pattern disappears!!
⇒ The observation of the experiment perturbes the physical system
and influences the result.
As soon as light is turned off, the interference pattern appears again.
⋆ As far as there is an interference pattern, we do not know through which
slit the electron went through.
This is an inherent property of quantum mechanics and can be understood in terms of
the Heisenberg uncertainty principle.
Heisenberg uncertainty principle
We assume that we observe the electrons with light, whose wavelength λL ∼ h/pL is large.
Through scattering with the electron, part of the momentum of light will be transferred
to the electron.
Let’s consider that L is big and the distance between slits d is small. Then, the momentum
of the electron px when the light hits it while passing through one of the slits will be
changed by:
1.4. DISCRETE STATES
15
∆px ∼ pL ∼ h/λL .
(1.16)
d > λL
(1.17)
In order that we can see (light source) through which slit the electron travels, the distance
between slits d should be bigger than λL :
on the other hand, d defines the uncertainty ∆x with which we can determine the position
of the particle. Then, combining the two previous equations we get:
∆px ∆x > h.
(1.18)
what is a manifestation of the Heisenberg uncertainty principle, i.e. the accurate position
and momentum of a particle cannot be measured at the same time. This implies that if we
increase the accuracy with which, for instance, the position is measured then momentum
will be defined accordingly with less accuracy. Position and momentum are conjugate
variables. We will do a rigorous derivation of this principle in the next chapters.
1.4
1.4.1
Discrete states
Quantization of spin/angular momentum
In 1921/22 Otto Stern and Walther Gerlach performed the following experiment at the
university of Frankfurt. They considered a collimated beam of silver atoms directed
between the poles of a magnet. This magnet produces an inhomogeneous field. As the
atoms are neutral, they do not experience a Lorentz force. However, due to their intrinsic
magnetic moments µ the atoms are deflected by the inhomogeneous field:
F = −∇(−µ.B)
(1.19)
The field causes an actual separation in space of atoms with spin up and spin down. The
magnetic moment of a silver atom is that of the spin of an electron.
This experiment is a manifestation of the discretization of the intrinsic angular momentum
of the electron, the spin s = h̄/2 with ms = ±1/2.
Both the spin and the angular momentum L of a particle are quantized.
By application of sequential Stern-Gerlach experiments, it was proven that in quantum
mechanics we cannot determine simultaneously two spin components, f.i. Sz and Sx .
Mathematically it is expressed by the nonconmutation of the respective operators.
1.4.2
Energy quantization
J. Balmer showed in 1885 that the Hydrogen atoms do not absorb or emit energy continuously, but only certain radiation wavelengths λ are observed:
1
1
1
= RH
− 2 ,
2
λ
n
m
(1.20)
16
CHAPTER 1. BASICS IN QUANTUM MECHANICS
where RH is the Rydberg constant:
RH = 109677, 6 cm−1
and
n, m ∈ N
with
m ≥ n + 1.
Within the classical explanation of the nature of an atom, i. e., that the electrons follow
Kepler orbits around the nucleus, one cannot explain this result. Since the electrons would
loose energy continuously through radiation, and that would mean
(i) that one should observe a continuous spectrum and
(ii) that the atoms are unstable.
Bohr (1913) provided the first explanation of the discrete spectral lines through two
postulates. His atomic theory founded the basis for the ”old quantum theory”.
Postulate 1 : The electron moves in circular orbits around the nucleus which are restricted by the orbital momentum:
m v r = n h/2π,
where n ∈ N
(1.21)
In these orbits of special radius, the electron does not radiate energy, as expected from
Maxwell’s laws. These orbits are called stationary states, and this is called the Bohr’s
quantization rule.
Postulate 2 : The energy of the atom has a definite value in a stationary orbit. The
electron can jump from one stationary orbit to another. If the electron jumps from an
orbit of higher energy E2 to an orbit of lower energy E1 , it emits a phonon, and the
wave/-length λ of the emitted radiation is given by the Einstein - Planck equation:
hc
.
(1.22)
λ
Accordingly, the electron can also absorb energy from some source and jump from E1 to E2
with E1 < E2 . The experimentally observed series-formulae consist of energy conditions
of the form:
E2 − E1 = hν =
hν = En − Em ;
En = −
RH h
.
n2
(1.23)
(1.24)