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Quantization as Selection Problem. Axiomatic Derivation of the Schrödinger Equation from Classical Mechanics
Siemens AG, previously Max Born Institute Berlin
1.Curiosities from Classical and Quantum Mechanics - What We Should Wonder About
2.Classical Mechanics after Euler - The Representation Bohr Had in Mind
3.The Conservation of Energy after Helmholtz – The Ground State Within Classical Mechanics
4.Quantization as Selection problem I. Derivation of the Stationary Schrödinger Equation
5.Quantization as Selection Problem II. Alternative Method for Solving the Stationary Schrödinger Equation
6.Derivation of the Time-Dependent Schrödinger Equation using Euler’s Principles of State Change
7.Prospects on Further Less Known Features of CM: Many-Particle Theory and Statistics
8.Summary – On the Interpretation of Non-Relativistic Quantum Mechanics
Dieter Suisky, Humboldt University at Berlin
D. Suisky & P. , Leibniz’ foundation of mechanics and the development of 18th century mechanics initiated by Euler, Proc. VII Intern. Leibniz Congress, Berlin 2001
P. & D. Suisky, On the selection problem in classical mechanics and in quantum mechanics, Nova Acta Leopoldina (2003) Suppl. 18 (in press)
D. Suisky & P. , On the derivation and solution of the Schrödinger equation. Quantization as selection problem, Proc. 5th Int. Symp. Frontiers of Fundamental Physics, Hyderabad (India), Jan. 8-11, 2003 (in press)
P. & D. Suisky, Quantization as selection problem, Int. J. Theor. Phys. (submit.)
P. & D. Suisky, Introduction into Quantum Field Theory of Solids. With an Elementary Introduction From Classical Mechanics to Quantum Field Theory (in prep.)
Curiosities in CM and QM . 4
1The Original Tasks and Goals of QM as Atom Mechanics Suggest to Introduce QM by means of Analyzing the Energy Conservation Law and Its Representation. 4
2Quantum Mechanics Needs Its Limit Case Classical Mechanics for Explanation/Foundation. 5
3Einstein (1907) : Quantization As Selection Problem6
4Three Requirements by Schrödinger (1926) to Any Quantization. 7
5Strengths and Weaknesses of Newtons’ Axioms. 8
6Internal and external Parameter should be separated. i=√(-1) may play physical role in CM . 9
7Intrinsic Limitation of Classical Motion: Possible and Impossible Configurations. 11
Classical Mechanics after Euler12
8Euler’s Axioms Fix only the Conservation of States -> Variability for Generalizations. 13
9Example for the Possibilities of Generalizations of the Principles of State Change : Selection Problems in CM. I. Relativistic vs Non-Relativistic Mechanics. 14
10Eulers Method of Maxima and Minima – Internal and External System Parameters. 15
11Euler’s Principles of State Change for Classical Bodies. 16
12The Euler’s Principles of State Change Can Freely Be Applied to Conservative Systems. 17
The Conservation of Energy after Helmholtz. 18
13Each System Exhibits a Ground State. 19
14Leibniz’s Principle, or: It is Not Necessary to Derive the Mechanical Energy Conservation Law from the Newtonean Equation of Motion. 20
15Relation Total Energy <–> Extension. 21
Quantization as Selection Problem I. Novel Derivation of the Stationary Schrödinger Equation. 22
16Selection Problems in CM. II. Newtonean vs Non-Newtonean CM . 23
17Selection Problems in CM. III. Classical vs Non-Classical Mechanics vs Non-Mechanics. 24
18Schematical Representations of the Transition CM -> QM . 25
19Systems with Unbounded (Momentum) Configurations. 27
20Non-classical Representation of the Energy. 29
21The Stationary Schrödinger Equation. 31
Quantization as Selection Problem II. Alternative Method for Solving the Stationary Schrödinger equation. 33
22Preparation: Rationalization of the Stationary Schrödinger Equation for the Harmonic Oscillator34
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23Selection of the Mathematically Distinguished States (E/hw half-integer)35
24Selection of the Physically Distinguished States (E/hw half-integer positive)37
Derivation of the Time-Dependent Schrödinger Equation. 38
25Transformation of Euler’s Principles of State Change to Quantum Systems. 38
26The Equation of State Change. 39
27The Equation of Motion, or: The Time-Dependent Schrödinger Equation. 40
28An Alternative: The Invariant Expression in the Non-Stationary Case. 41
Prospects to Further „Unknown“ Contents of CM: Many-Particle Theory and Statistics. 42
29‛By-produkt’ Permutation Symmetrie of the Wave Function. 42
30Which (In)Distinguishability is Important for Statistics?. 44
Summary – On the Interpretation of Non-Relativistic Quantum Mechanics. 46
Curiosities in CM and QM
1The Original Tasks and Goals of QM as Atom Mechanics Suggest to Introduce QM by means of Analyzing the Energy Conservation Law and Its Representation
1)Explanation of the stability of the atoms
2)Explanation of the discreteness of the atomic spectra
3)Calculation of the intensities of the spectral lines
That are stationary properties -> Question: It is appropriate to introduce QM by means of equations of motion? (Newton’s equation of motion, d’Alembert’s wave equation, the Hamilton-Jacobi equation etc.)
All know Bohr’s postulates (1913) – more lasting are Bohr’s prerequisites (1913):
1)The principles of state conservation in CM and QM are the „same“ (energy and other conserved quantities);
2)The principles of state change in CM and QM are different.
All variables, entities, notions not being independent of orbital motion can be transferred from CM to QM.
Examples: Energy, configuration, (effective) extension of a system
What else to learn from Bohr’s atom model? (i) It contains only internalsystem parameters; the energy spectrum does not depend on external parameters, such as initial conditions. (ii) Energy and extension (radius of orbit) are related to each other as ordered sets (Bohr’s orbits to be replaced with the maxima of electron density or the like).
2Quantum Mechanics Needs Its Limit Case Classical Mechanics for Explanation/Foundation
Generally accepted statement:
“The formulation of the fundamental laws of quantum theory is on principle impossible without evoking classical mechanics. . Thus, quantum mechanics assumes a very strange position among the physical theories: It contains classical mechanics as limith case and, at the same time, needs this limith case for its own foundation.”
(L. D. Landau & E. M. Lifschitz, Quantum mechanics, 1966, pp.2f.)
Has CM got all necessary means for deriving QM without additional requirements / assumptions?
Our answer: yes!
3Einstein (1907) : Quantization As Selection Problem of Possible States
„From the aforegoing it is clear, in which sense the molecular-kinetic theory of heat has to be modified, in order to be brought into harmony with the distribution law of black-body radiation. While until now, one thought the molecular motions to be subject to exactly the same lawfulnesses which hold true for the motions of the bodies of our sensory world, we are now forced . to make the assumption, that the mannifold of the states, which they are able to assume, is less than for the bodies ofour [everyday] experience.“
Systems are to be described and characterized primarily through their (stationary) energetic states.
The set of possible stationary states determines the set of possible state changes / motions.
Note: Euler’s contributions to physics are overshadowed by that to mathematics; Cauchy’s contributions to physics are overshadowed by that to mathematics; Einstein’s contributions to QM are overshadowed by that to theory of relaticity;
4Three Requirements by Schrödinger (1926) to Any Quantization
1.The „quantum equation” should carry „the quantum conditions in itself“, -> to manage without boundary conditions;
2.There should be special mathematical solution methods for differential equations of the type of the stationary Schrödinger equation accounting for the non-classical character of the quantization problem; therefore, they should be different from the classical methods of calculating the eigenmodes of strings, resonators etc.;
3.The derivation should uniquely result in that the energy and not the frequency values become discrete.
Quantization is fundamentally different from classical discretization as observed for standing waves
Our derivation of the stationary Schrödinger equation will obey these requirements.
5Strengths and Weaknesses of Newtons’ Axioms (Please, Teach + Discuss the Original Wording!)
CM º Newtonian mechanics? Newton’s axioms = „the (only) axioms“ for point mechanics?
(N1)“Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.”
(N2)“A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.”
(N3)“To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each orr are always equal and always opposite in direction.”
Advantage: Solid basis for all tasks which of point mechanics, especially, for planetary motions (Newton’s original goal!)
Disadvantage: Not only the conservation of states, but also the modus and way of their change are axiomatically fixed.
=> Generalizations are difficult, last but not least conceptionally and methodically.
Here: Euler’s representation of CM
Addendum: Rest and motion:
Axiom (N1) supposes
Application: Behaviour of Hamilton-Jacobi equation w.r.t. Möbius transformation
6Internal and external Parameters should be separated. i=√(-1) may play a physical role in CM
Standard solution for the harmonic oscillator ‛mixes’ internal and externalparameters:
Separation of the dynamical variables:
contains only internal parameters and the time; describes rotations in phase space; group property (Huygens’s principle was first formulated in point mechanics!):
(Chapman-Kolmogorov equation) – Eigenvectors of as new dynamical variables:
Most simple time-dependencies:
They obey not equations of motion of 2nd order, such as and , but of 1storder
This is a factorization of the equation of motion of 2nd order: .
=> two ‛trivial‛ first integrals of the oscillator motion:
, the variables factorize the Hamilton function:
ØSeparation of internal motion and external ‛influences’ leads to valueable insights;
Øthis separation is possible only when using .
The fundamental equations of motion are always of 1storder in time. Their propagation kernels obey the Chapman-Kolmogorov equation (Huygens’s principle)
7Intrinsic Limitation of Classical Motion: Possible and Impossible Configurations
In state E the work storage, A, of a system is bounded ():
=> Conservative system can assume not all, but onlycertainconfigurations, x : :
Are there systems, for which this limitation does not apply?
Classical Mechanics after Euler
All neccessary means for deriving non-classical-mechanical basic equations are contained within CM – though these means are widely unknown.
Euler’s representation of CM is as easy to understand as Newton’s one. However: From it, it is a much smaller step to the ‛not understandable’ non-classical theories. -> It can help to overcome the defensive attitude ‛I don’t understand this anyway’ (which I had earlier, too!).
8Euler’s Axioms Fix only the Conservation of States -> Variability for Generalizations
Euler’s axioms contain onlythe internal principles, the principles of state conservation; they are formulated for the interaction-free body:
E0)An interaction-free body is either in the state at rest, or in the state of straight uniform motion.
E1)A body who stays at rest at one time will remain forever at rest, until he is disturbed by an external cause and set in motion.
E2)A body remains in the state of straight uniform motion with constant velocity and direction, until no external cause changes this state.
Euler’s axioms apply cum granis solis to other areas than point mechanics
Note: These axioms carry the logical relationship (XOR) being characteristic for the relationship CM<->QM, CM<->STR, …(?) (physics cannot contradict logics!)
9Example for the Possibilities of Generalizations of the Principles of State Change : Selection Problems in CM. I. Special-Relativistic vs Non-Relativistic Mechanics
(B1) Either the state change is independent of the state itself -> non-relativistic mechanics,
(B2) Or the state change is not independent of the current state -> special-relativistic mechanics.
10Eulers Method of Maxima and Minima – Internal and External System Parameters
QM: Atomic electronic states are strukturally stable against not too large disturbations (J. Franck & G. Hertz 1914). CM: Kepler orbits are strukturally instable.
=> QM needs a representation of energy not as a continuous function of continuous external variables (such as the initial conditions), but of the internal system parameters only, like in Bohr’s atom model.
The current state of a system is determined by both, internal and external parameters.
Important exception: State at rest: Total energy and extension are determined uniquely through V=VminandT=Tmin. =>
Hypothesis 1: Parameters, the values of which are determined through the condition V=VminandT=Tmin, are internal parameters.
Hypothesis 2: States, in which the system’s extension in (momentum) space belongs to extremalous values of appropriate functions, are internal states.
For which systems all states are internal states?
11Euler’s Principles of State Change for Classical Bodies
Up to order dt:
(PK1)The changes of the conserved quantities (dZ=dv) depend onlyon the external causes (F) (mediated through the external transformation Uext), but not on the non-conserved quantities (x); in particular, dv=0, if F=0;
(PK2)The changes of the conserved quantities (dZ) are independent of the conserved quantities (Z) themselves;
(PK3)The changes of the non-conserved quantities (dx) depend onlyon the conserved quantities (Z) (mediated through the internal transformation Uint); the external causes (F) influence the non-conserved quantities (x) only indirectly via the conserved quantities (Z);
(PK5)As soon as the external causes vanishe, the body remains in the state it has assumed at this time: for , if for .
12Euler’s Principles of State Change Can Freely Be Applied to Conservative Systems
(PS1)The changes of the conserved quantities (dZ) depend only on the external causes (), but not on the internal causes (H0); in particular, dZ=0, if ;
(PS2)The changes the conserved quantities (dZ) are independent of the conserved quantities (Z) themselves;
(PS3)The changes of the non-conserved quantities (dx, dp) depend only on the conserved quantities (Z); the external causes () influence the non-conserved quantities (x, p) only indirectly via the conserved quantities (Z);
(PS4)The changes of the conserved (dZ) and of the non-conserved quantities (dx, dp) are independent upon each other;
(PS5)As soon as the external causes vanishe, the system remains in the state it has assumed at this time: for , if for .
=> Hamilton’s equations of motion,
The Conservation of Energy after Helmholtz
(Lectures on the Dynamics of Discrete Mass Points)
13Each System Exhibits a Ground State
„We start from the assumption, that it is impossible, through any combination of natural bodies, to permanently create moving force out off nothing.“
(H. Helmholtz, On the Conservation of Force, 1847)
14Leibniz’s Principle, Or: It is Not Necessary to Derive the Mechanical Energy Conservation Law from Newton’s Equation of Motion
„If an arbitrary number of movable mass points moves only under the influence of such forces with which they act against themselves or which are directed against fixed centres, then, the sum of all living forces is at all that times the same, in which all points assume the same relative positions against each other and against the possibly existing centres, whatever their orbits and velocities have intermediately been.“
„Namely, if the sum of the living forces for each constellation of the masses, which is repeated during the motion, gets also the same value, then, this sum should, although being composed only from the masses and the squares of the velocities, be a pure function of the coordinates of the mass points.“
15Relation Total Energy <–> Extension
A conservative system increases / decreases its characteristic dimensions, if its total energy increases / decreases.
Example linear harmonic oscillator:
Note: This holds true also in Bohr’s atom model!
Quantization as Selection Problem I. Novel Derivation of the Stationary Schrödinger Equation
„We are faced here with the full force of the logical opposition between an
either – or (point mechanics)
both – and (wave mechanics)
This would not matter much, if the old system were to be dropped entirely and to be replaced by the new.“
(Erwin Schrödinger, Nobel Award speech 1933)
Question: Is there a relationship
Logics <-> Selection (principles) ?
16Selection Problems in CM. II. Newtonean vs Non-Newtonean CM
‛Newton’s’ equation of motion:
Without contradiction to Newton’s 1st and 3rd axioms, one can set/require/postulate
=> rather than
-> Classical, but Non-Newtonean mechanics.
Relaxing Newton’s 2nd axiom opens ways for generalization
1) Helmholtzean condition for the definition of classical quantities: is semi-definit:
2) Helmholtzean condition for the definition of nonclassical quantities: is indefinit: both equations,
are allowed; nouniform force law for both domains => noclassical (connected with orbits and forces) equations of motion for the total domain
3) Non-mechanical case:
No mechanics possible
-> Hierarchy of selection problems:
1.Newtonean CM with vs Non-Newtonean CM with ;
2.CM (either – or) vs Non-CM (and);
3.Mechanics vs Non-Mechanics.
Alternative definitions of the set of possible (momentum) configurations are possible without coming into conflict with the axiomatics.
Logical relation in the selection problems (xor) is the same as that between Euler’s axioms E1 and E2 (see Axiom E0) => the alternatives above exclude each other and are mutually dependent, are, in their manner, „in harmony to each other”, too.
Selection problem (2) is a generalization of the Bohr-Heisenberg complementarity of the relationship CM <-> QM.
Conventional scheme CM -> QFT
The wave-particle dualism of classical (!) physics is brought in quantum physics
Derivation of the Schrödinger equation. Right side: after Schrödinger; left side: this approach
19Systems with Unbounded (Momentum) Configurations
2 tightly interrelated results suggest a natural and axiomatic modification of CM towards QM:
Result1: Relaxing the division of the set of all (momentum) configurations into possible and impossible;
Result2: Representation of extension and energy as internal parameters for allE-values, not only for (state at rest)
The fundamental differences between CM and Non-CM:
1. The notion of orbit as a point-wise relation between space and momentum coordinates losses its sense:
ØThere is no unique algebraic relation like between them;
ØThere is no common parametrization like , ;
2. There are no longer coordinates , describing the system’s boundaries, because and are no longer (semi-)definit.
1. All (momentum) configurations are to be considered together, all configurations are connected with all momentum configurations;