Physical Measurement. Uncertainty Principle for Measurement

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1 Physical Measurement Uncertainty Principle for Measurement Measuring rod is marked in equispaced intervals of which there are N of one unit of measurement size of the interval is 1/N The measuring variable is denoted t Small increment in Δt of the variable t will be detected, if t and t+ Δt lie on the opposite sides of the division t and t+ Δt do not fall within the same measuring interval 1

2 t and t+ Δt do not fall within the same measuring interval Δt 1/N --> NΔt 1 Let ΔN=1/2N half width of the range of the number N ΔNΔt 1/2 Suppose we think about Δt as representing time N denotes the number of equal subintervals into which the unit of time has been partitioned be a measuring apparatus (frequency) Trade-off between the accuracy of measurement and the range of variability of the measuring instrument Uncertainty of the observed value and the uncertainty of the frequency range to perform the measurement are related If one is small, the other has to be large 2

the Fourier transform of x(t) the inverse Fourier transform of X(f) X( f ) = x(t) " e #2$itf dt x(t) = #% t stands for time, f stands for frequency, and x

3 At what times (or time intervals), do these frequency components occur? FT gives the spectral content of the signal, but it gives no information regarding where in time those spectral components appear! the Fourier transform of x(t) the inverse Fourier transform of X(f) X( f ) = x(t) " e #2$itf dt x(t) = #% t stands for time, f stands for frequency, and x denotes the signal x denotes the signal in time domain and the X denotes the signal in frequency domain % & % & #% X( f ) " e 2$itf df The signal x(t), is multiplied with an exponential term, at some certain frequency "f", and then integrated over ALL TIMES! 3

There is only a minor difference between STFT and FT In STFT, the signal is divided into small enough segments, where these segments (portions) of the signal can be assumed to be stationary For this

4 There is only a minor difference between STFT and FT In STFT, the signal is divided into small enough segments, where these segments (portions) of the signal can be assumed to be stationary For this purpose, a window function "w" is chosen The width of this window must be equal to the segment of the signal where its stationarity is valid... Narrow window good time resolution, poor frequency resolution 4

Width window good frequency resolution, poor time resolution A Fourier transform of f is a function of frequency v Let be Δv the frequency range It can be proved that ΔtΔv 1/4 π If Δt is small, f

5 Width window good frequency resolution, poor time resolution A Fourier transform of f is a function of frequency v Let be Δv the frequency range It can be proved that ΔtΔv 1/4 π If Δt is small, f corresponds to a small interval of the whole function Lower frequencies are not represented For higher frequency range, a big interval of the function Good frequency representation, bad time resolution, Δt is big 5

6 One cannot know what spectral components exist at what instances of times What one can know are the time intervals in which certain band of frequencies exist, which is a resolution problem The problem has to do with the width of the window function that is used Narrow window good time resolution, poor frequency resolution Wide window good frequency resolution, poor time resolution Conservation of Information Let the first measurement limit the value m by t 1 < m < t 2 and let t=t 2 -t 1 Let the second measurement refine the first one by t 1 < m < t 2 and let t =t 2-t 1 The information gained is I=log 2 ((t 2 -t 1 )/(t 2-t 1))=log 2 ( t/ t ) 6

7 Measurement which is accurate to within t implies the use of measurement frequencies N that satisfies the uncertainty relation ΔNΔt 1/2 and ΔN Δt 1/2 If the measurment are made in the most economical way, we will have ΔNΔt =1/2 and ΔN Δt =1/2 Δt/ Δt = ΔN / ΔN I(Δt,Δt )=log 2 (Δt,Δt )=-log 2 (ΔN/ ΔN ) I(Δt,Δt )+ I(ΔN/ ΔN )=0 Information about one variable is gained at the expense of an equal loss of information about the other variable Principle of conservation of information 7

8 Quantum Mechanics Build on Max Planck s discovery in 1900 that certain physical quantities and energy can only change in discrete jumps Physical measurements have an intrinsically probabilistic character: Repetitions of an observation using the same experimental apparatus with the same initial conditions will in general yield different measurements of the observed variable Totality of large observations is governed by statistical laws The wavefunction in quantum mechanics evolves according to the Schrödinger equation into a linear superposition of different states It describes the probability of the presence of certain states The actual measurements always find the physical system in a definite state 8

9 Wavefunction is a function from a space that consists of the possible states of the system into the complex numbers The laws of quantum mechanics (i.e. the Schrödinger equation) describe how the wave function evolves over time The values of the wave function are probability amplitudes complex numbers the squares of the absolute values of which, give the probability distribution that the system will be in any of the possible states Any future evolution is based on the state the system was discovered to be in when the measurement was made The measurement "did something" to the process under examination Whatever that "something" may be does not appear to be explained by the basic theory 9

The best known example is the "paradox" of the Schrödinger's cat a cat is apparently evolving into a linear superposition of basis vectors that can be characterized as an "alive cat" and states that

10 The best known example is the "paradox" of the Schrödinger's cat a cat is apparently evolving into a linear superposition of basis vectors that can be characterized as an "alive cat" and states that can be described as a "dead cat" Each of these possibilities is associated with a specific nonzero probability amplitude; the cat seems to be in a "mixed" state (suppose probability to be alive and o be death are equals) However, a single particular observation of the cat does not measure the probabilities: it always finds either an alive cat, or a dead cat After that measurement the cat stays alive or dead 10

11 The question is: how are the probabilities converted to an actual, sharply welldefined outcome? Different interpretations of quantum mechanics propose different solutions of the measurement problem The Copenhagen interpretation is rooted in the philosophical positivism It claims that quantum mechanics deals only with the probabilities of observable quantities; all other questions are considered "unscientific" (metaphysical). Copenhagen regards the wavefunction as a mathematical tool used in the calculation of probabilities with no physical existence (not an element of reality) Waveform collapse is therefore a meaningless concept; the waveform only describes a specific experiment 11

12 Consciousness causes collapse proposes that the presence of a conscious being causes the wavefunction to collapse However, this interpretation depends on a definition of "consciousness". This interpretation, however, is seen as unscientific by critics due to its inability to be falsified Conjugate pairs Another unexpected property of the nature: Physical variables come in conjugate pairs Position and momentum Energy and time Both of which cannot be simultaneously measured with accuracy 12

13 Uncertainty principle If p and q are conjugate pairs Position and momentum Energy and time Both cannot be simultaneously measured with arbitrarily high accuracy Uncertainty principle A Fourier transform of f is a function of frequency v Let be Δv the frequency range ΔtΔv 1/4 π Planck: Energy is bounded in discrete packets called quanta Every energy quantum is associated with an oscillatory phenomenon, having a certain frequency 13

14 When a physical system passes from one energy state to another, it either emits or absorbs a quantum of energy whose magnitude E is equal to the difference of the two states Energy difference in terms of the difference v in frequency of the two state E= hv h=6.62*10-27 erg-second is universal constant of the nature (Panck s constant) We can solve for v and we get: ΔtΔE h/4 π Heisenberg uncertainty principle for time and energy We can not have unlimited simultaneous precision in the measurement of both time and energy 14

15 Uncertainty Principle According to the principles of special relativity, space and time have an equal standing There is a way of to transform the coordinates from time into space The space coordinates x, y, z bear the corresponding momentum variables p x, p y, p z (momentum is the product of the mass and velocity of an object) ΔxΔp x h/4 π ΔyΔp y h/4 π ΔzΔp z h/4 π Momentums are represented by Wavefunctions Inequalities govern the accuracy of a measurement of space coordinates and their corresponding momenta Arise from the property of nature, and from mathematical description of measurement without to properties governed by physical law, see ΔtΔv 1/4 π 15

16 Conservation of Information We measure energy of a photon of light emitted by it Error of measurement lies within some limits E 1 Similar measurement of time when the particle of light was emitted must be an amount t 1 such that the inequality Δt 1 ΔE 1 h/4 π holds E 1 t 1 should be as small as possible Second observation of emitted energy is made with the purpose of refining our knowledge with E 2 and t 2 Information gain log 2 ( E 1 / E 2 ) Since E 1 t 1 = E 2 t 2 log 2 ( E 1 / E 2 )+log 2 ( t 1 / t 2 ) 16

17 Information gain of one of the conjugate variables is balanced b information loss of the other variable In situation that are suboptimal, the information gained by a measurement will be less than the information lost Second law of thermodynamics Physical measure of entropy The difference in the quantity of information between two states of a physical system is equal to the negative of the corresponding difference in entropy of the two states We are concerned with information transfer from one system to another Information transfer are the results of certain changes of states 17

18 Each physical change of state, such as the firing of a neuron is accompanied by the release of certain amount of energy E The corresponding change of information I is given by E= c I c is a constant which is proportional to the temperature T c=k T log 2 k is Boltzmann s constant I measured in bits E= (k T log 2) I=(9.57*10-17 )T I= ergs Enormous number of bits of information will be needed to produce an energy change which can be sensed at the macroscopic level 18

19 Change of 1 bit of information content requires the expenditure of at least kt log 2 ergs of energy It takes more energy to produce a bit at high temperature than at low Ordinary room temperature it is approximately 3*10-14 ergs, much less than it is released by a typical chemical reaction of a single atom Electronics, around kt ergs per operation Information processing as a heat engine 19

Uncertainty Principle

Uncertainty Principle n n A Fourier transform of f is a function of frequency v Let be Δv the frequency range n It can be proved that ΔtΔv 1/(4 π) n n If Δt is small, f corresponds to a small interval