Class 17: The Uncertainty Principle

Transcription

1 Class 7: The Uncertainty Princile Wae ackets and uncertainty A natural question to ask is if articles behae like waes, what is waing? The arying quantity is related to the robability of finding the article at a articular location A wae of a well-defined single waelength will extend eerywhere and is not localized To localize a article the wae must hae a finite extent This leads to the concet of a wae acket, which is shown schematically below Δx The width of the wae acket, Δx, is a measure of the uncertainty in the osition of the article As the wae acket moes, the wae inside also moes and not necessarily at the same seed as the eneloe of the wae acket Hence the number of maxima in length Δx will change Suose we try to determine the waelength associated with the wae acket by counting the maxima We find The uncertainty in the waelength from our counting is Tyically n Using equation (7) to eliminate n, we get x λ = (7) n x λ = n (7) n λ x λ (73) The de Broglie relations relate momentum to waelength Hence an uncertainty in the waelength gies an uncertainty in the momentum This can be written as h h = λ (74) λ x x h (75)

2 By considering the relationshi between energy and frequency, the uncertainty in energy of a wae acket localized to a time interal Δt is such that E t h (76) Relations like that in equations (75) and (76) were first encountered by Werner Heisenberg in 97, and hence are referred to as the Heisenberg uncertainty rincile From a more recise analysis, the uncertainty rincile is x ħ, (77) and E t ħ (78) Consequences of the uncertainty rincile Consider a article of mass m confined in a -D box of length L Since x L, the momentum of the article cannot be exactly zero Hence the article has a minimum kinetic energy K = m ħ 8mL (79) The mechanical energy of a harmonic oscillator of mass m and sring constant k is E = + kx (70) m Heisenberg s uncertainty rincile sets a non-zero lower limit to the mechanical energy which can be found by making the change of ariables so that ( ) ( ) 4 4 = km q, x = km y, (7) k E = q + y m ( ) (7) The constraint from the uncertainty rincile is q y ħ (73) Assuming that and x are normally distributed, we hae

3 k ( ) ( ) k E = q + y = q + y m m, (74) because the mean alues of and x are both zero From equation (74), we see that a oint ( q, y) of energy E lies on a circle The constraint (73) requires the oint to lie in a region bounded by a hyerbola The minimum energy occurs when the circle just touches the hyerbola It does this on the line q = y Hence the minimum allowed energy is E min k = ħ = ħ ω, (75) m where ω is the angular frequency of the oscillator These examles show that a article confined by a otential has a minimum zero-oint energy Some consequences of this zero-oint energy are that a crystalline solid will melt if comressed to high density een at zero temerature, and cold fusion could occur if the density was sufficiently large that the zero oint energy is high enough for enetration of the Coulomb barrier to occur More detailed treatment of wae ackets A harmonic wae in D can be reresented by the functional form y i( kx ωt ) = Re Ae, (76) where k and ω take definite alues Clearly this wae function extends eerywhere Since a wae acket is to reresent a localized article, it cannot hae a single waelength A general exression to describe a wae acket is ( ) i kx ω( k ) t y = Re A k e dk (77) The relationshi between ω and k is called the disersion relation For examle for waes on a string the disersion relation is ω = kc, where c is the seed of roagation of the wae Suose that the amlitude function A(k) is uniform oer a narrow range of wae numbers (, + k ) and zero outside that range Then Let k = + κ, so that + y = Re A e dk i kx ω( k ) t (78) 3

4 y = Re A e d 0 i x+ κ x ω( + κ ) t κ (79) Because k is small, we can exand ω ( k 0 + κ ) in a Taylor series and discard second and higher order terms We get i x+ κ x ω( ) t ω ( ) κt y = Re A e dκ 0 i( x ω0t) iκ ( x ω0 t) = Re Ae e dκ, 0 (70) where ω0 = ω ( ), and ω = ω ( k ) 0 0 We see that the wae acket consists of a harmonic wae function by an eneloe function ( ω0 ) 0 iκ ( x ω0 t) F x t e dκ = ω0 i( x ω0t) multilied f x t = Ae The harmonic wae roagates at seed ω0 = (7) k 0 This is called the hase elocity of the wae acket The eneloe roagates at a seed g = ( ) (7) dk This is called the grou elocity of the wae acket For waes on a string, since ω = kc, we see that the hase elocity is equal to the grou elocity Phase and grou elocity for non-relatiistic and relatiistic free articles Assume that the aroriate exression to use for the energy in the de Broglie relation is the mechanical energy For a free article of mass m, Using the de Broglie relations, this becomes E = K = (73) m 4

5 ( k ) The disersion relation for the waes associated with a free article is ħ ω = ħ (74) m We see that the hase and grou elocities are ω = ħ m k (75) g ω ħk = = =, k m m ħk = = =, dk m m (76) and that it is the grou elocity that gies the classical relationshi between momentum and elocity This is consistent with the idea that it is the eneloe of the wae acket that localizes the article Next assume that the aroriate energy to use in the de Broglie relations is the total relatiistic energy For a free article of mass m, we hae The de Broglie relations gie The disersion relation is ( ) ( ) E = c + mc (77) ( ω ) = ( c k ) + ( mc ) ħ ħ (78) mc ω = k c + ħ (79) The grou elocity is k k ħk c g = c c c dk = ω dk = ω = ħω = E (730) Again, we see that it is the grou elocity that is equal to the elocity of the article 5