Period 3-interpretations .(1) Copenhagen(orthogonal)interpretations 1920s-1930s. (2)Additional-variable interpretations. de Broglie(1926), Bohm (1952) o 3) Relative-state interpretations. Everett (1957).(many-worlds interpretation) .(4)Modal interpretations. van Fraassen( 1973) .(5)Consistent-histories interpretations Griffiths(1984) .( 6)Physical collapse models. Pearle(1976)
Period 3 – interpretations ⚫ (1) Copenhagen (orthogonal) interpretations, 1920s-1930s. ⚫ (2) Additional-variable interpretations. de Broglie (1926), Bohm (1952). ⚫ (3) Relative-state interpretations. Everett (1957). (many-worlds interpretation). ⚫ (4) Modal interpretations. van Fraassen (1973). ⚫ (5) Consistent-histories interpretations. Griffiths (1984). ⚫ (6) Physical collapse models. Pearle (1976)
lI Formalism of quantum mechanics Axiom I(ref. BG03 1. Every physical system S is associated to a Hilbert space H; the physical states of s are represented by normalized vectors(called statevectors)u> of H. Physical observables O of the system are represented by self- adjoint operators in H; the possible outcomes of a measurement ofo are given by its eigenvalues On Olon>=On?' The eigenvalues on are real and the eigenvectors on> form a complete orthonormal set in the hilbert space H
II. Formalism of quantum mechanics – Axiom I (ref.BG03) ⚫ 1. Every physical system S is associated to a Hilbert space H; the physical states of S are represented by normalized vectors (called “statevectors”) |ψ> of H. Physical observables O of the system are represented by selfadjoint operators in H; the possible outcomes of a measurement of O are given by its eigenvalues on , ⚫ O|on> = on |on>. ⚫ The eigenvalues on are real and the eigenvectors |on> form a complete orthonormal set in the Hilbert space H
Axiom Il 2. To determine the state y(to)> of the system S at a given initial time to, a complete set of commuting observables for s is measured the initial statevector is then the unique common eigenstate of such observables. Its subsequent time evolution is governed by the Schrodinger equation indy(t>ldt= Hy(t> o which uniquely determines the state at any time once one knows it at the initial time the operator H is the Hamiltonian of the system S
Axiom II ⚫ 2. To determine the state |ψ(t0 )> of the system S at a given initial time t0 , a complete set of commuting observables for S is measured: the initial statevector is then the unique common eigenstate of such observables. Its subsequent time evolution is governed by the Schrödinger equation ⚫ iℏd|ψ(t)>/dt = H|ψ(t)>, ⚫ which uniquely determines the state at any time once one knows it at the initial time. The operator H is the Hamiltonian of the system S
AxiomsⅢ and V 3. The probability of getting, in a measurement at time t, the eigenvalue on in a measurement of the observable o is given by P(on=k<only(t)>2, where u(t)> is the state of the system at the time in which the measurement is performed 4. The effect of a measurement on the system S is to drastically change its statevector from 四f)>to|on>:|ψ(t)>( before measurement) 0n>(after measurement. This is the famous postulate of wavepacket reduction(or collapse of state vector)
Axioms III and VI ⚫ 3. The probability of getting, in a measurement at time t, the eigenvalue on in a measurement of the observable O is given by P(on )=|<on |ψ(t)>|2 , where |ψ(t)> is the state of the system at the time in which the measurement is performed. ⚫ 4. The effect of a measurement on the system S is to drastically change its statevector from |ψ(t)> to |on>: |ψ(t)> (before measurement) → |on> (after measurement). This is the famous postulate of wavepacket reduction (or collapse of state vector)
Two other quantization methods Feynmans path integral path e iSIh ∑ where S=Ldt is quantization method action of a path Stochastic quantization Classical particles subject to method random diffusion
Two other quantization methods Feynman’s path integral quantization method Stochastic quantization method ∑path e iS/ℏ where S=∫Ldt is action of a path Classical particles subject to random diffusion