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Uncertainty Relations as Consequence of the Lorentz Transformations

VADIM N. MATVEEV, O.V. Matveev

Published Dec 1, 2011• 1 citation
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Abstract

It is believed that the uncertainty relation of momentum and coordinates, as well as the uncertainty relation of energy and time in practice are not observed in the macrocosm. The objective of the work at hand was to demonstrate the existence of general physical uncertainty relations that extend to macrobodies. A macroscopic object consisting of a rod equipped with a pair of synchronized clocks and a macroscopic object in and of itself performing the function of an ideal physical clock are examined. General physical relations are directly derived from Lorentz transformations for the case of the object's one-dimensional motion (along the X axis) – the uncertainty relation of the object's x coordinate and the projection of its impulse along the X axis, px, and the uncertainty relation of the object's observation time, t, and its energy, E. The relations take the form: ΔpxΔx ≥ H and ΔEΔt ≥ H. The H value in the relation has action dimensions and is dependent upon the precision of the object's clocks and/or upon the properties of the physical clock. Despite the interpretation of the concept of uncertainty being different from that in quantum mechanics, the relations derived in the limiting case with ideal physical clock take the form of ΔpxΔx ≥ h and ΔEΔt ≥ h, where h is the Planck constant.

Related papers

Study on Uncertainty Relations as a Consequence of the Lorentz Transformations

VADIM N. MATVEEV, Oleg V. Matvejev

A macroscopic object equipped with synchronized clocks is examined. General physical relations are directly derived from Lorentz transformations for the case of one-dimensional motion (along the X axis) – the uncertainty relation of the object's x coordinate and the projection of its impulse along the X axis, px, and the uncertainty relation of the object's observation time, t, and its energy, E. The relations take the form: \(\Delta\)px\(\Delta\)x \(\ge\) H and \(\Delta\)E\(\Delta\)t \(\ge\) H. The H value in the relation has action dimensions and is dependent upon the precision of the rod's clocks and its mass. It is shown that if the macroscopic object in and of itself performs the function of an ideal physical clock, the relations derived in the limiting case then take the form of \(\Delta\)px\(\Delta\)x \(\ge\) h and \(\Delta\)E\(\Delta\)t \(\ge\) h, where h is the Planck constant.

Book Publisher International (a part of SCIENCEDOMAIN International)•Aug 11, 2021•0 citations

Quantum Dynamical Probability and the Uncertainty Principle

Francesco R. Ruggeri

The uncertainty principle (linked to p,x or E,t) plays an important role in quantum mechanics. For example, an electron with an exact momentum p is described by a wavefunction exp(ipx). This function exists at all x points and so it is suggested in the literature that there is complete uncertainty as to the electron’s position. (More specifically P(x)=exp(-ipx)exp(ipx) normalization constant=constant and so all x points have the same probability.) Thus if p is completely certain, it is said that position x is completely unknown. We note, however, that exp(ipx) consists of cos(px) and sin(px) which are repeated probability distributions (with zero endpoints). Thus even though they exist in all of space, due to repetition which in turn is due to spatial invariance, there is a specific length number, namely the wavelength. Thus we argue that in classical physics there is a set of points (x,p), but in quantum mechanics ( p , x with its dx interval), with the latter being equal to hbar/p. Then the only uncertainty is within the dx interval i.e. the wavelength and spatial density is linked to momentum i.e. there is dynamical probability. In particular, there is a frequency hbar/E and a wavelength hbar/p which define an internal clock and ruler and the two are independent. Thus within a wavelength one has a probabilistic picture (in order to have endpoints defining the dx interval) and no sense of time, yet on average time exists because of hbar/E and the particle moves in space with a constant v. This begs the question: How do time and space x arise in such a scenario? We speculate the following. Assume there exists a property called energy which defines an object. This object, however, may break into two different objects. As a result, one introduces a parameter called time to distinguish between the two systems (E) and (E1,E2). For a certain time range one has a single object, while for a later time interval, one has two objects. Each of the two objects in the second system, however, needs to be distinguished as well and so another parameter, namely space x is used for this. If the two objects existed at the same x and t, how would one know there were two unless another parameter were introduced? We use t and x as the distinguishing parameters and furthermore give these a measure i.e. dt and dx. We then argue that from these ideas one may show the notion of speed and direction (because there are two objects) as well as conservation of momentum arises. Given the notion of an interval, we argue it must be described by a probability distribution and we link this to constant/p and exp(ipx). We finally note that since spatial density is linked to dynamical probability, one must consider motion associated with a density which may lead even to collective motion.

•Dec 21, 2022•0 citations

Exploring the implications of the uncertainty relationships in quantum mechanics

Huai‐Yu Wang

Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.

Frontiers in Physics•Dec 8, 2022•3 citations

Flow of time during energy measurements and the resulting time-energy uncertainty relations

Ismael L. Paiva, Augusto César Lobo, Eliahu Cohen

Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.

arXiv:2106.00523•Quantum•Apr 7, 2022•16 citations

Uncertainty Principles for Position- and Momentum-Like Operators

Lara Ortelli, Gael Silva

At the quantum scale, objects such as fundamental particles, atoms, and molecules are modeled as localized wavepackets rather than concrete objects. As such, some quantities that are well-defined at our human scale physics instead become uncertain. For instance, quantum objects do not have a precise position, and instead their position is understood probabilistically and we look to the expected, or average, value of the position. Another quirk of quantum mechanics is that quantities such as position and momentum become operators that act on functions, similar to matrices acting on vectors. As such, quantities that may have commuted classically may not at the quantum level. For instance, position (x) and momentum (p) commute in classical physics (xp = px), but they do not commute at the quantum level (xp ≠ px). For noncommuting operators A and B, the uncertainty principle qualitatively describes how well one can know A and B simultaneously: the more certain one is of the value of A, the less one knows the value B and vice versa. The standard example is that of position and momentum: the Heisenberg uncertainty principle states that the more certain one is of the position of a quantum object, the more uncertain one is about its momentum. The classic uncertainty principles between noncommuting operators are those of position and momentum, the z and x or y components of angular momentum, and number and phase operators. These operators are very rigid and do not provide easy generalization to more general operators. In this project, we used an infinite matrix representation to analyze the uncertainty principle underlying position and momentum which allows for broader generalization. The biggest component of this project was determining eigenvalues of infinite banded matrices which was aided by the use of the mathematical software Mathematica.

•Mar 27, 2026•0 citations

PRINCIPLES OF SPECIAL RELATIVITY

Mircea S. Rogalski, Stuart B. Palmer

Ψ∂ (4.1) The harmonic trial solution of this equation with respect to a fixed inertial frame ( is given by )tx, ( ) ( ) ( )ctxikectxktx −Ψ=Ψ−Ψ=Ψ 00 orcos, In another inertial frame( ),, tx ′′ considered to be in uniform translational motion with velocity as described by the Galilean transformation (2.13), with respect to the fixed frame, the wave is represented by a wave function ,v ( ) ( ), related to ,x t x′ ′Ψ Ψ t as 2 2i.e., x t x x x t x x x x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂ Ψ ∂ Ψ= + = =′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ and 2 2i.e., 2 x t v v v t x t t t x t x tt x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂Ψ ∂ Ψ ∂ Ψ ∂ Ψ ∂ Ψ= + = − + = − +′ ′ ′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2t∂ ∂ ∂ where use has been made of known properties of partial derivatives. It follows that obeys the following wave equation, with respect to the moving inertial frame ( tx ′′Ψ , ) ( ) tx v tx vc ′∂′∂ Ψ∂−′∂ Ψ∂=′∂ Ψ∂− 2 22 2 (4.2) which is obviously different from the wave equation (4.1) that holds in the fixed inertial frame. In other words the electromagnetic wave equation is not invariant with respect to a uniform translation of the inertial frame, using a Galilean transformation. Inserting into Eq.(4.2) harmonic solution written as we obtain 2 2 2 2 2 2( ) ( ) 2 ( ) i.e., orc v k k w v k w w v c w c− − = − − = − ± = ± v (4.3) Hence Galileo's transformation predicts that the velocity of light should be different from c, if measured with respect to a moving inertial frame. We therefore have a choice to make before we can fully understand the electromagnetic phenomena. We can either accept Galileo's principle of relativity which states that all inertial frames are equivalent or the Galilean transformation (2.13). We can sacrifice Galileo's principle of relativity by saying that there is a privileged inertial frame, called the ether, in which it is supposed that the electromagnetic wave velocity is c as given by Maxwell's equations, although they made no essential reference to the ether. A long series of investigations has been carried out, in order to detect some effect of the motion of the earth through the ether, always with negative results. All of the experiments (the best known being that of Michelson-Morley) have indicated that the speed of light is always the same in all directions and is independent of the relative motion of observer, source and transmitting medium. Einstein has reaffirmed Galileo's principle of relativity on the equivalence of all the inertial frames and, in addition, has assumed as a basic postulate the experimental fact that the speed of light is always constant. As a consequence, the Galilean transformation had to be replaced by another, in order to preserve the same speed of light in all systems. Einstein's basic assumptions are known as the postulates of special relativity: 1. The laws of physics take the same form in all inertial systems 2. The speed of light in empty space is a universal constant c and is the same for all observers. It is independent of the state of motion of the emitting body Special relativity is only concerned with transformations between inertial frames, as indicated by the restriction contained in the first postulate. General relativity covers reference frames in arbitrary relative motion, and is based on the principle of equivalence, confirmed experimentally, which unifies gravitational and inertial forces. The first of Einstein's postulates implies that it is only possible, by means of physical measurements, to demonstrate that two gravity-free reference frames are moving relative to each other. There is no physical basis to infer that any such frame is intrinsically at rest or in uniform motion. The notion of absolute rest is not consistent with the first postulate. The second postulate is in contradiction with the Galilean transformation which would preserve the form of Newton's laws of motion, as illustrated by Eq.(4.3). Therefore, we must find a transformation between two frames moving with uniform relative motion, which is consistent with the second postulate. We must then generalize the laws of Newtonian mechanics in a form which is invariant under that transformation, as required by the first postulate. A fundamental requirement of special relativity is that the laws of Newtonian mechanics are valid in the low speed limit and must be obtained as limiting cases of relativistic formulae for .cv << This assumption plays, for special relativity, a role analogous to that of Bohr's correspondence principle in quantum mechanics (see Chapter 28). 4.2. THE LORENTZ TRANSFORMATION Einstein's procedure is to locate an event in a reference frame by its spatial Cartesian coordinates x, y and z and a temporal coordinate t. Consider two frames S and S' moving with uniform relative motion, such that the Cartesian axis system in coincides with that in S, Oxyz, at time ,S ′ zyxO ′′′′ .0==′ tt As illustrated in Figure 4.1, the x and axes are parallel to the direction of relative motion: in moves in the x direction with velocity v, and in x′ , the frameS ′S S, the frameS ′ moves with velocity in the direction. A pulse of light emitted from O at time v− x′ 0=t spreads as a spherical wave travelling with speed c, so that the equation of the wave front in S, at a time t will be (4.4) 22222 tczyx =++ 0 0' z' y' x,x' z y Figure 4.1. Cartesian axes of two inertial frames in uniform relative motion The second postulate requires that the wave front in S ′ can be represented by (4.5) 22222 tczyx ′=′+′+′ In other words an observer in also sees a spherical wave travelling from its origin if not, any change of the wave front could be used to infer that the system is moving uniformly instead of being at rest. The invariance of the expression S ′ ,O′ S ′ (4.6) 2222222222 tczyxtczyx ′−′+′+′=−++ is a basic requirement that the transformation between the two systems must satisfy. The Galilean transformation (2.13), which in our case reads ttzzyyvtxx =′=′=′−=′ ,,, is not consistent with the invariance property (4.6), as it gives xvttvtczyxtczyx 2222222222222 −+−++=′−′+′+′ The first postulate brings some simplification to the possible form of the desired transformation. Since a uniform rectilinear motion in S must go over into a uniform rectilinear motion in S', the transformation must be linear. A nonlinear transformation would predict acceleration in S' even if the velocity were constant in S. The choice of the origins in space and time has avoided constant terms, and the transformation will be homogenous. The directions perpendicular to the relative motion, which are effectively at rest, must be left unchanged by the transformation. We then take the general form BtAxx +=′ yy =′ (4.7) zz =′ FtDxt +=′ Further simplification comes from the argument that the origin O', which has the coordinate moves along the x axis with velocity ,0=′x ,/ vdtdx = that is ABvBtAx /or0 −=+= while the origin O, with coordinate ,0=x moves along x-axis with velocity Recalling that when , we can write .'/' vdtdx −= 0=x 0=t 00 ort D B Bx A B t v F F F ′ − ⋅′ ′= ⋅ + = − = so that Inserting now Eqs.(4.7) into Eq.(4.5), we obtain the equation .FA = ( ) ( ) 02 22222222222222222 =−+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −−++−=′−′+′+′ DcBAxt c BAtczyxDcAtczyx The identity (4.6) requires that 2222 =−=−=− DcB c BADcA Since ,B vA= we obtain 2222 /1 /, /1 , /1 cv cvD cv vB cv A − −= − −= − = (4.8) so that Eqs.(4.7) now read 22 /1 cv vtxx − −=′ yy =′ (4.9) zz =′ /1 / cv cvxtt − −=′ The four equations (4.9) are known as the Lorentz transformation, which in this case applies to the relative motion of inertial frames along the x-axis. It is a straightforward matter to show that Eqs.(4.9) are consistent with the invariance relation (4.6), which is thus a sufficient as well as a necessary condition for the Lorentz transformation. With the usual abbreviations ( )1 /1 1, 222 ≥ − = − == γ β γβ cvc v (4.10) the Lorentz transformation simplifies to ( ) ( )x x ct x x ctγ β γ β′ ′= − = + ′ a) yyyy ′==′ or b) (4.11) zzzz ′==′ c) ⎟⎠ ⎞⎜⎝ ⎛ ′+′=⎟⎠ ⎞⎜⎝ ⎛ −=′ x c ttx c tt βγβγ d) The right hand side shows the inverse transformations where the primed and unprimed quantities are interchanged and the sign of the relative velocity is reversed, because its direction is the only difference between the systems S and S'. In the low velocity limit the Lorentz transformation reduces to the Galilean transformation since the only difference consists in the presence of the factor γ and the term ,1< that, in Eq.(4.11a), γ indicates a change of measuring stick length and in Eq.(4.11d) a change in the clock rate in the Lorentz transformation. The term ( )xc/β− in Eq.(4.11d) shows that the time is dependent on the position of a given event. Therefore, the immediate consequences of the Lorentz transformation require the revision of the usual concepts of simultaneity, time and length. Simultaneity Consider two events occuring at the same time ttt == 21 at two different positions and in the unprimed system S. According to Eqs.(4.11) the events are recorded in the primed system S' at different times 1x 2x / /, 1 1 t x c t xt t cβ ββ β − −′ ′= = − − so that the apparent time interval in S' is given by ( ) 0 1 / 2 21 12 ≠− −=′−′=′∆ β β cxx ttt (4.12) and hence, it is a function of the relative velocity v. The events will not appear as simultaneous to observers in S'. We are then forced to abandon the intuitive idea of absolute simultaneity, which is based on our experience with objects of low velocity, and accept the relativity of simultaneity, which depends on the coordinate system and applies in the domain of high velocities. Length contraction Consider a rod at rest in the primed system S' lying along the x-axis and having the length 12 xx ′−′=λ given by the distance between its ends 21 and xx ′′ measured at the same instant of time t'. The length in an inertial frame in which the rod is at rest is called its proper length. To define length in a frame S in which the rod is moving, an observer must locate the position of both end points, 21 and xx ′′ , at the same time From the inverse equations (4.11) we obtain 1 2 .t t t= = 1 1 2 1 x ct x ct x xx x x xβ ββ β β − −′ ′ ′ ′= = − = − − − − so that the apparent length 12 xxl −= is 21 βλ −=l (4.13) This equation is known as the Lorentz contraction and states that a rod parallel to the direction of its motion is shortened by the factor .1 2β− It results from the fact that simultaneous events in the rest frame of one rod are not simultaneous in the rest frame of the other. A rod transverse to its direction of motion is unchanged in length. The proper length λ is independent of the orientation of the rod, as a manifestation of the isotropy of inertial rest frame S'. If the orientation of a rod is neither longitudinal nor transverse to the direction of relative motion, both the length and the inclination are changed in the laboratory frame S, as an effect of a contraction in the direction of the relative motion with no change of transverse dimensions.

CRC Press eBooks•Jan 8, 2014•0 citations
It is believed that the uncertainty relation of momentum and coordinates, as well as the uncertainty relation of energy and time in practice are not observed in the macrocosm. The objective of the work at hand was to demonstrate the existence of general physical uncertainty relations that extend to macrobodies. A macroscopic object consisting of a rod equipped with a pair of synchronized clocks and a macroscopic object in and of itself performing the function of an ideal physical clock are examined. General physical relations are directly derived from Lorentz transformations for the case of the object's one-dimensional motion (along the X axis) – the uncertainty relation of the object's x coordinate and the projection of its impulse along the X axis, px, and the uncertainty relation of the object's observation time, t, and its energy, E. The relations take the form: ΔpxΔx ≥ H and ΔEΔt ≥ H. The H value in the relation has action dimensions and is dependent upon the precision of the object's clocks and/or upon the properties of the physical clock. Despite the interpretation of the concept of uncertainty being different from that in quantum mechanics, the relations derived in the limiting case with ideal physical clock take the form of ΔpxΔx ≥ h and ΔEΔt ≥ h, where h is the Planck constant.
A macroscopic object equipped with synchronized clocks is examined. General physical relations are directly derived from Lorentz transformations for the case of one-dimensional motion (along the X axis) – the uncertainty relation of the object's x coordinate and the projection of its impulse along the X axis, px, and the uncertainty relation of the object's observation time, t, and its energy, E. The relations take the form: Δ\DeltaΔpxΔ\DeltaΔx ≥\ge≥ H and Δ\DeltaΔEΔ\DeltaΔt ≥\ge≥ H. The H value in the relation has action dimensions and is dependent upon the precision of the rod's clocks and its mass. It is shown that if the macroscopic object in and of itself performs the function of an ideal physical clock, the relations derived in the limiting case then take the form of Δ\DeltaΔpxΔ\DeltaΔx ≥\ge≥ h and Δ\DeltaΔEΔ\DeltaΔt ≥\ge≥ h, where h is the Planck constant.
The uncertainty principle (linked to p,x or E,t) plays an important role in quantum mechanics. For example, an electron with an exact momentum p is described by a wavefunction exp(ipx). This function exists at all x points and so it is suggested in the literature that there is complete uncertainty as to the electron’s position. (More specifically P(x)=exp(-ipx)exp(ipx) normalization constant=constant and so all x points have the same probability.) Thus if p is completely certain, it is said that position x is completely unknown. We note, however, that exp(ipx) consists of cos(px) and sin(px) which are repeated probability distributions (with zero endpoints). Thus even though they exist in all of space, due to repetition which in turn is due to spatial invariance, there is a specific length number, namely the wavelength. Thus we argue that in classical physics there is a set of points (x,p), but in quantum mechanics ( p , x with its dx interval), with the latter being equal to hbar/p. Then the only uncertainty is within the dx interval i.e. the wavelength and spatial density is linked to momentum i.e. there is dynamical probability. In particular, there is a frequency hbar/E and a wavelength hbar/p which define an internal clock and ruler and the two are independent. Thus within a wavelength one has a probabilistic picture (in order to have endpoints defining the dx interval) and no sense of time, yet on average time exists because of hbar/E and the particle moves in space with a constant v. This begs the question: How do time and space x arise in such a scenario? We speculate the following. Assume there exists a property called energy which defines an object. This object, however, may break into two different objects. As a result, one introduces a parameter called time to distinguish between the two systems (E) and (E1,E2). For a certain time range one has a single object, while for a later time interval, one has two objects. Each of the two objects in the second system, however, needs to be distinguished as well and so another parameter, namely space x is used for this. If the two objects existed at the same x and t, how would one know there were two unless another parameter were introduced? We use t and x as the distinguishing parameters and furthermore give these a measure i.e. dt and dx. We then argue that from these ideas one may show the notion of speed and direction (because there are two objects) as well as conservation of momentum arises. Given the notion of an interval, we argue it must be described by a probability distribution and we link this to constant/p and exp(ipx). We finally note that since spatial density is linked to dynamical probability, one must consider motion associated with a density which may lead even to collective motion.
Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.
At the quantum scale, objects such as fundamental particles, atoms, and molecules are modeled as localized wavepackets rather than concrete objects. As such, some quantities that are well-defined at our human scale physics instead become uncertain. For instance, quantum objects do not have a precise position, and instead their position is understood probabilistically and we look to the expected, or average, value of the position. Another quirk of quantum mechanics is that quantities such as position and momentum become operators that act on functions, similar to matrices acting on vectors. As such, quantities that may have commuted classically may not at the quantum level. For instance, position (x) and momentum (p) commute in classical physics (xp = px), but they do not commute at the quantum level (xp ≠ px). For noncommuting operators A and B, the uncertainty principle qualitatively describes how well one can know A and B simultaneously: the more certain one is of the value of A, the less one knows the value B and vice versa. The standard example is that of position and momentum: the Heisenberg uncertainty principle states that the more certain one is of the position of a quantum object, the more uncertain one is about its momentum. The classic uncertainty principles between noncommuting operators are those of position and momentum, the z and x or y components of angular momentum, and number and phase operators. These operators are very rigid and do not provide easy generalization to more general operators. In this project, we used an infinite matrix representation to analyze the uncertainty principle underlying position and momentum which allows for broader generalization. The biggest component of this project was determining eigenvalues of infinite banded matrices which was aided by the use of the mathematical software Mathematica.
Ψ∂ (4.1) The harmonic trial solution of this equation with respect to a fixed inertial frame ( is given by )tx, ( ) ( ) ( )ctxikectxktx −Ψ=Ψ−Ψ=Ψ 00 orcos, In another inertial frame( ),, tx ′′ considered to be in uniform translational motion with velocity as described by the Galilean transformation (2.13), with respect to the fixed frame, the wave is represented by a wave function ,v ( ) ( ), related to ,x t x′ ′Ψ Ψ t as 2 2i.e., x t x x x t x x x x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂ Ψ ∂ Ψ= + = =′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ and 2 2i.e., 2 x t v v v t x t t t x t x tt x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂Ψ ∂ Ψ ∂ Ψ ∂ Ψ ∂ Ψ= + = − + = − +′ ′ ′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2t∂ ∂ ∂ where use has been made of known properties of partial derivatives. It follows that obeys the following wave equation, with respect to the moving inertial frame ( tx ′′Ψ , ) ( ) tx v tx vc ′∂′∂ Ψ∂−′∂ Ψ∂=′∂ Ψ∂− 2 22 2 (4.2) which is obviously different from the wave equation (4.1) that holds in the fixed inertial frame. In other words the electromagnetic wave equation is not invariant with respect to a uniform translation of the inertial frame, using a Galilean transformation. Inserting into Eq.(4.2) harmonic solution written as we obtain 2 2 2 2 2 2( ) ( ) 2 ( ) i.e., orc v k k w v k w w v c w c− − = − − = − ± = ± v (4.3) Hence Galileo's transformation predicts that the velocity of light should be different from c, if measured with respect to a moving inertial frame. We therefore have a choice to make before we can fully understand the electromagnetic phenomena. We can either accept Galileo's principle of relativity which states that all inertial frames are equivalent or the Galilean transformation (2.13). We can sacrifice Galileo's principle of relativity by saying that there is a privileged inertial frame, called the ether, in which it is supposed that the electromagnetic wave velocity is c as given by Maxwell's equations, although they made no essential reference to the ether. A long series of investigations has been carried out, in order to detect some effect of the motion of the earth through the ether, always with negative results. All of the experiments (the best known being that of Michelson-Morley) have indicated that the speed of light is always the same in all directions and is independent of the relative motion of observer, source and transmitting medium. Einstein has reaffirmed Galileo's principle of relativity on the equivalence of all the inertial frames and, in addition, has assumed as a basic postulate the experimental fact that the speed of light is always constant. As a consequence, the Galilean transformation had to be replaced by another, in order to preserve the same speed of light in all systems. Einstein's basic assumptions are known as the postulates of special relativity: 1. The laws of physics take the same form in all inertial systems 2. The speed of light in empty space is a universal constant c and is the same for all observers. It is independent of the state of motion of the emitting body Special relativity is only concerned with transformations between inertial frames, as indicated by the restriction contained in the first postulate. General relativity covers reference frames in arbitrary relative motion, and is based on the principle of equivalence, confirmed experimentally, which unifies gravitational and inertial forces. The first of Einstein's postulates implies that it is only possible, by means of physical measurements, to demonstrate that two gravity-free reference frames are moving relative to each other. There is no physical basis to infer that any such frame is intrinsically at rest or in uniform motion. The notion of absolute rest is not consistent with the first postulate. The second postulate is in contradiction with the Galilean transformation which would preserve the form of Newton's laws of motion, as illustrated by Eq.(4.3). Therefore, we must find a transformation between two frames moving with uniform relative motion, which is consistent with the second postulate. We must then generalize the laws of Newtonian mechanics in a form which is invariant under that transformation, as required by the first postulate. A fundamental requirement of special relativity is that the laws of Newtonian mechanics are valid in the low speed limit and must be obtained as limiting cases of relativistic formulae for .cv << This assumption plays, for special relativity, a role analogous to that of Bohr's correspondence principle in quantum mechanics (see Chapter 28). 4.2. THE LORENTZ TRANSFORMATION Einstein's procedure is to locate an event in a reference frame by its spatial Cartesian coordinates x, y and z and a temporal coordinate t. Consider two frames S and S' moving with uniform relative motion, such that the Cartesian axis system in coincides with that in S, Oxyz, at time ,S ′ zyxO ′′′′ .0==′ tt As illustrated in Figure 4.1, the x and axes are parallel to the direction of relative motion: in moves in the x direction with velocity v, and in x′ , the frameS ′S S, the frameS ′ moves with velocity in the direction. A pulse of light emitted from O at time v− x′ 0=t spreads as a spherical wave travelling with speed c, so that the equation of the wave front in S, at a time t will be (4.4) 22222 tczyx =++ 0 0' z' y' x,x' z y Figure 4.1. Cartesian axes of two inertial frames in uniform relative motion The second postulate requires that the wave front in S ′ can be represented by (4.5) 22222 tczyx ′=′+′+′ In other words an observer in also sees a spherical wave travelling from its origin if not, any change of the wave front could be used to infer that the system is moving uniformly instead of being at rest. The invariance of the expression S ′ ,O′ S ′ (4.6) 2222222222 tczyxtczyx ′−′+′+′=−++ is a basic requirement that the transformation between the two systems must satisfy. The Galilean transformation (2.13), which in our case reads ttzzyyvtxx =′=′=′−=′ ,,, is not consistent with the invariance property (4.6), as it gives xvttvtczyxtczyx 2222222222222 −+−++=′−′+′+′ The first postulate brings some simplification to the possible form of the desired transformation. Since a uniform rectilinear motion in S must go over into a uniform rectilinear motion in S', the transformation must be linear. A nonlinear transformation would predict acceleration in S' even if the velocity were constant in S. The choice of the origins in space and time has avoided constant terms, and the transformation will be homogenous. The directions perpendicular to the relative motion, which are effectively at rest, must be left unchanged by the transformation. We then take the general form BtAxx +=′ yy =′ (4.7) zz =′ FtDxt +=′ Further simplification comes from the argument that the origin O', which has the coordinate moves along the x axis with velocity ,0=′x ,/ vdtdx = that is ABvBtAx /or0 −=+= while the origin O, with coordinate ,0=x moves along x-axis with velocity Recalling that when , we can write .'/' vdtdx −= 0=x 0=t 00 ort D B Bx A B t v F F F ′ − ⋅′ ′= ⋅ + = − = so that Inserting now Eqs.(4.7) into Eq.(4.5), we obtain the equation .FA = ( ) ( ) 02 22222222222222222 =−+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −−++−=′−′+′+′ DcBAxt c BAtczyxDcAtczyx The identity (4.6) requires that 2222 =−=−=− DcB c BADcA Since ,B vA= we obtain 2222 /1 /, /1 , /1 cv cvD cv vB cv A − −= − −= − = (4.8) so that Eqs.(4.7) now read 22 /1 cv vtxx − −=′ yy =′ (4.9) zz =′ /1 / cv cvxtt − −=′ The four equations (4.9) are known as the Lorentz transformation, which in this case applies to the relative motion of inertial frames along the x-axis. It is a straightforward matter to show that Eqs.(4.9) are consistent with the invariance relation (4.6), which is thus a sufficient as well as a necessary condition for the Lorentz transformation. With the usual abbreviations ( )1 /1 1, 222 ≥ − = − == γ β γβ cvc v (4.10) the Lorentz transformation simplifies to ( ) ( )x x ct x x ctγ β γ β′ ′= − = + ′ a) yyyy ′==′ or b) (4.11) zzzz ′==′ c) ⎟⎠ ⎞⎜⎝ ⎛ ′+′=⎟⎠ ⎞⎜⎝ ⎛ −=′ x c ttx c tt βγβγ d) The right hand side shows the inverse transformations where the primed and unprimed quantities are interchanged and the sign of the relative velocity is reversed, because its direction is the only difference between the systems S and S'. In the low velocity limit the Lorentz transformation reduces to the Galilean transformation since the only difference consists in the presence of the factor γ and the term ,1< that, in Eq.(4.11a), γ indicates a change of measuring stick length and in Eq.(4.11d) a change in the clock rate in the Lorentz transformation. The term ( )xc/β− in Eq.(4.11d) shows that the time is dependent on the position of a given event. Therefore, the immediate consequences of the Lorentz transformation require the revision of the usual concepts of simultaneity, time and length. Simultaneity Consider two events occuring at the same time ttt == 21 at two different positions and in the unprimed system S. According to Eqs.(4.11) the events are recorded in the primed system S' at different times 1x 2x / /, 1 1 t x c t xt t cβ ββ β − −′ ′= = − − so that the apparent time interval in S' is given by ( ) 0 1 / 2 21 12 ≠− −=′−′=′∆ β β cxx ttt (4.12) and hence, it is a function of the relative velocity v. The events will not appear as simultaneous to observers in S'. We are then forced to abandon the intuitive idea of absolute simultaneity, which is based on our experience with objects of low velocity, and accept the relativity of simultaneity, which depends on the coordinate system and applies in the domain of high velocities. Length contraction Consider a rod at rest in the primed system S' lying along the x-axis and having the length 12 xx ′−′=λ given by the distance between its ends 21 and xx ′′ measured at the same instant of time t'. The length in an inertial frame in which the rod is at rest is called its proper length. To define length in a frame S in which the rod is moving, an observer must locate the position of both end points, 21 and xx ′′ , at the same time From the inverse equations (4.11) we obtain 1 2 .t t t= = 1 1 2 1 x ct x ct x xx x x xβ ββ β β − −′ ′ ′ ′= = − = − − − − so that the apparent length 12 xxl −= is 21 βλ −=l (4.13) This equation is known as the Lorentz contraction and states that a rod parallel to the direction of its motion is shortened by the factor .1 2β− It results from the fact that simultaneous events in the rest frame of one rod are not simultaneous in the rest frame of the other. A rod transverse to its direction of motion is unchanged in length. The proper length λ is independent of the orientation of the rod, as a manifestation of the isotropy of inertial rest frame S'. If the orientation of a rod is neither longitudinal nor transverse to the direction of relative motion, both the length and the inclination are changed in the laboratory frame S, as an effect of a contraction in the direction of the relative motion with no change of transverse dimensions.