The Uncertainty Principle and Complementarity
The pivotal advancements in quantum theory were marked by de Broglie's hypothesis of matter waves, Heisenberg's formulation of matrix mechanics, and Schrodinger's wave equation, which interconnected these concepts Heisenberg's uncertainty principle and Bohr's foundational discussions further solidified the early development of quantum theory, culminating in a significant phase of its evolution.
The theory addresses the longstanding issue of wave-particle duality in both light and matter, providing a comprehensive and accurate explanation of the associated phenomena.
This solution requires sacrificing the objective treatment of physical phenomena, which involves relinquishing the classical understanding of space-time and causality This traditional framework relies on our capacity to distinctly separate the observer from the observed.
The challenges of using both photon and wave concepts of light are evident when examining a monochromatic point source in front of a diffraction grating Assuming the grating has infinite resolving power, the wave theory indicates that the diffracted light can only reach specific, defined points.
Key texts in the development of quantum theory include W Heisenberg's "The Physical Principles of the Quantum Theory" (1949), N Bohr's "Atomic Theory and the Description of Nature" (1961), and L de Broglie's "Introduction to the Study of Wave Mechanics" (1930) Additionally, the Solvay Congress of 1927 played a significant role in shaping quantum discussions, while E Schrödinger's lectures on wave mechanics further contributed to the field's understanding.
2 L de Broglie, Ann d Phys (10) 3, 22 (1925), (Thesis, Paris, 1924); cf also A Einstein, Berl Ber
3 W Heisenberg, Z Physik 33, 879 (1925): cf also M Born and P Jordan, Z Physik 34, 858 (1925); M Born, W Heisenberg and P Jordan, Z Physik 35, 557 (1926); P.A.M Dirac, Proc Roy Soc Lond 109, 642 (1925)
4 E Schrodinger, Ann d Phys (4) 79,361,489,734 (1926); 80, 437 (1926); 81, 109 (1926), Col- lected Papers on Wave Mechanics, Blackie and Son Ltd., London (1928)
* Schrodinger followed de Broglie's idea of matter waves in setting up his equation Later he proved the equivalence of his approach to that of Heisenberg's See ref 4 above
6 N Bohr, Naturwiss 16, 245 (1928) (Also printed as Appendix II in A.a.N.) t i.e phenomena concerning Nuclear, Atomic and Molecular Physics www.pdfgrip.com
Quantum mechanics operates on the principle that light waves emerging from a diffraction grating exhibit a path difference equivalent to an integral number of wavelengths, supporting the superposition principle validated by extensive experimental data This phenomenon is observable even at low intensities typical of individual atom emissions From a particle perspective, the process involves an atom emitting light, followed by scattering at the grating and subsequent absorption The directionality of diffracted photons, calculable through wave theory, necessitates the presence of all atoms in the grating Attempting to pinpoint where a photon strikes the grating without affecting the diffraction pattern introduces significant challenges, as the behavior of photons relies on the positions of all atoms, rendering classical wave field theories insufficient for predicting their statistical behavior.
It is impossible to create a wave field that has both of the following properties: (i) its intensity is zero at all points of the grating except for one specific line, and (ii) only certain directions of the diffracted rays are present To adhere to the superposition principle, we must assume that when a photon interacts with a specific line of the grating, the other lines do not affect the resulting diffraction pattern Consequently, the diffraction pattern observed must be identical to that produced by a photon interacting with only that single line of the grating.
The requirement discussed is applicable to all interference experiments, not just a specific type of diffraction experiment These experiments share a common feature: light waves from the same source travel different paths, resulting in a phase difference before converging at another point It is essential to postulate that if a photon takes one of these paths, the interference pattern predicted by wave theory cannot be observed.
As already mentioq.ed, this requirement is contained in another requirement which is more general and which can be formulated in the following way: All the properties
The behavior of a photon can be uniquely characterized by the wave field corresponding to a specific measurement, which can be generated through a superposition of plane waves with varying directions and wavelengths This concept leads to the definition of a wave-packet, allowing us to represent the photon's presence in a particular space-time region Even without analyzing measurement results, we can indicate the photon's location by referencing the significant wave amplitudes of its associated wave-packet.
Sec 1 The Uncertainty Principle and Complementarity 3 different from zero only within the concerned space-time region We denote the (complex) phase of a plane wave by
The wave number vector \( k \), with components \( k_i' \), indicates the direction of the wave normal and has a magnitude of \( \frac{2\pi}{\lambda} \), where \( \lambda \) represents the wavelength The angular frequency \( \omega \), defined as \( 2\pi \) times the oscillation frequency \( v \), is a function of the components \( k_1, k_2, k_3 \) and is uniquely determined by the characteristics of the waves For electromagnetic waves in a vacuum, this relationship holds true.
The velocity of light in a vacuum, denoted as c, plays a crucial role in understanding wave fields Notably, the subsequent conclusions drawn are not reliant on the specific form of the function oo(kl, k2, k3) In fact, for the most general wave field, every component of any arbitrary field strength can be effectively represented.
In the analysis of the function A(k), it is established that if u(x, t) is non-zero only within a defined spatial region characterized by dimensions ~X1, ~X2, and ~X3, and simultaneously, A(k) remains non-zero only within the corresponding "k-space" region defined by dimensions ~k1, ~k2, and ~k3, then the products ~i ~ki, where i = 1, 2, 3, cannot be negligibly small Instead, these products must maintain a minimum order of 1.
We shall speak about the quantitative refinement of this principle and its proof later
An analogous principle applies to the time spread ~t, where the function u(-;, t) is non-zero at a specific spatial point (Xl, X2, x3) This is similar to the frequency spread ~oo, which is confined to a specific region of k-space where A(k) remains significantly non-zero.
The condition (1.4) indicates that when a wave-packet's width matches the spacing between two grating lines, the resulting angular width of the diffracted rays is sufficiently broad to encompass at least two consecutive diffraction maxima, leading to a blurred diffraction pattern.
The interaction of photons with matter allows us to derive insights about material bodies, highlighting the importance of conditions essential for maintaining a corpuscular perspective in interference phenomena To facilitate calculations regarding the exchange of energy and momentum between light and matter, the concept of a quantum of light is introduced, grounded in the strict adherence to the laws of conservation of momentum and energy This exchange is accurately represented by assigning the photon a momentum proportional to its direction of propagation and a corresponding energy defined by Planck's constant divided by 2π.
* The notation, 1; = h/21t is due to Dirac www.pdfgrip.com
4 General Principles of Quantum Mechanics Sec 1
Remembering the definition of the vector k and eq (1.2), this statement can be expressed by the relations
Many-Particle Interactions - Operator Calculus
Schrodinger Equation and Operator Calculus
3 The Wave Function of Free Particles
We now introduce the basic assumptions regarding the position probabilities
In the non-relativistic region, where particle velocities are significantly lower than the speed of light, the momentum probabilities W(p1, p2, p3) and the position probabilities W(x1, x2, x3) align with the uncertainty principles and the wave-like behavior of matter This relationship highlights the fundamental nature of particles in quantum mechanics.
By restricting ourselves to the 'non-relativistic case we are excluding the photons from consideration We shall consider the relativistic theory in Chapters IX and X
We imagine, therefore, the functions
The construction follows the framework outlined in (1.3), ensuring that Ik I and m meet the criteria specified in (1.8'), confirming their positivity The introduction of the factor 1/V(21t)3 will be elaborated upon later Additionally, we define the complex conjugate function tp.(x, t) as _1_ fA (k) e- i [(r~-CDt] d 3 k.
Now introducing, instead of k and m the momentum p = nr and the energy
E = 'lim of the particle according to I in (3.1) and (3.1 *), these functions can also be written as
The functions AcP) and A(t:) differ by a numerical factor such that www.pdfgrip.com
Schrodinger Equation and Operator Calculus
3 The Wave Function of Free Particles
We now introduce the basic assumptions regarding the position probabilities
The momentum probabilities W(p1, p2, p3) and the position probabilities W(x1, x2, x3) of a particle align with the uncertainty relations and the wave nature of matter This discussion focuses on the non-relativistic regime, where the particle's velocity is significantly lower than the speed of light, and the associated wave frequencies are connected to this context.
By restricting ourselves to the 'non-relativistic case we are excluding the photons from consideration We shall consider the relativistic theory in Chapters IX and X
We imagine, therefore, the functions
The construction follows the specifications outlined in (1.3), ensuring that Ik I and m meet the criteria set in (1.8'), thus remaining positive The introduction of the factor 1/V(21t)3 will be elaborated on later Additionally, we define the complex conjugate function tp.(x, t) as _1_ fA (k) e- i [(r~-CDt] d 3 k.
Now introducing, instead of k and m the momentum p = nr and the energy
E = 'lim of the particle according to I in (3.1) and (3.1 *), these functions can also be written as
The functions AcP) and A(t:) differ by a numerical factor such that www.pdfgrip.com
14 General Principles of Quantum Mechanics Sec 3
IAW/2dP 1 dP2dP3 = IA(k)12dk 1 dk2dk3 Setting
Then 'V and ",*can be written as , ) ;
(X t) = _ _ 1 -J (1:) e! (;-;) dip t p , V(2n A)I f/J 'P , tp* (x ,t) = 1 !tp.cP) e -, ~