5.1 Uncertainty principle and particle current

Transcription

1 5.1 Uncertainty principle and particle current Slides: Video Momentum, position, and the uncertainty principle Text reference: Quantum Mechanics for Scientists and Engineers Sections

2 Uncertainty principle and particle current Momentum, position and the uncertainty principle Quantum mechanics for scientists and engineers David Miller

3 Momentum and the momentum operator For momentum we write an operator ˆp We postulate this can be written as ˆp i with xo yo zo x y z where x o, y o, and z o are unit vectors in the x, y, and z directions

4 Momentum and the momentum operator With this postulated form ˆp i we find that pˆ m m and we have a correspondence between the classical notion of the energy E p E V m and the corresponding Hamiltonian operator of the Schrödinger equation ˆ pˆ H V V m m

5 Momentum and the momentum operator Note that pˆ expikriexpikr kexpikr This means the plane waves expikr are the eigenfunctions of the operator ˆp with eigenvalues k We can therefore say for these eigenstates that the momentum is p k Note that p is a vector, with three components with scalar values not an operator

6 Position and the position operator For the position operator the postulated operator is almost trivial when we are working with functions of position It is simply the position vector, r, itself At least when we are working in a representation that is in terms of position we therefore typically do not write ˆr though rigorously we should The operator for the z-component of position would, for example, also simply be z itself

7 The uncertainty principle Here we illustrate the position-momentum uncertainty principle by example We have looked at a Gaussian wavepacket before We could write this as a sum over waves of different k-values, with Gaussian weights or we could take the limit of that process by using an integration k k, exp exp G z t i k tkz dk k k

8 The uncertainty principle We could rewrite k k, exp exp G z t i k t kz dk k k at time t = 0 as where,0 exp z k ikz dk k k exp k k k k k

9 The uncertainty principle In k k k k k k exp k k k is the representation of the wavefunction in k space is the probability P k strictly, the probability density that if we measured the momentum of the particle actually the z component of momentum it would be found to have value k

10 The uncertainty principle With k k k k exp k then this probability (density) of finding a value for the momentum would be k k Pk kk exp k This Gaussian corresponds to the statistical Gaussian probability distribution with standard deviation k k

11 The uncertainty principle Note also that,0 exp z k ikz dk is the Fourier transform of and, as is well known the Fourier transform of a Gaussian is a Gaussian specifically here z,0 exp k z k k k k

12 The uncertainty principle If we want to rewrite z,0 exp k z in the standard form z z,0 exp z where the parameter z would now be the standard deviation in the probability distribution for z then kz 1/

13 The uncertainty principle From kz 1/ if we now multiply by to get the standard deviation we would measure in momentum we have pz which is the relation between the standard deviations we would see in measurements of position and measurements of momentum

14 The uncertainty principle This relation pz is as good as we can get for a Gaussian For example a Gaussian pulse will broaden in space as it propagates even though the range of k values remains the same

15 The uncertainty principle It also turns out that the Gaussian shape is the one with the minimum possible product of p and z So quite generally pz which is the uncertainty principle for position and momentum in one direction

16 The uncertainty principle in Fourier analysis Uncertainty principles are well known in Fourier analysis One cannot simultaneously have both a well defined frequency and a well defined time If a signal is a short pulse it is necessarily made up out of a range of frequencies 1 t The shorter the pulse is the larger the range of frequencies

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18 5.1 Uncertainty principle and particle current Slides: Video Particle current Text reference: Quantum Mechanics for Scientists and Engineers Section 3.14

19 Uncertainty principle and particle current Particle current Quantum mechanics for scientists and engineers David Miller

20 The divergence of a vector In Cartesian coordinates the divergence of a vector F is F F x y Fz divff x y z We can visualize this in terms of the flux F of some quantity such as mass or charge through a small cuboidal box of sides x, y, and z centered at some point x ( xo, yo, zo) z y (x o, y o, z o ) y z x

21 The divergence of a vector Because F represents the flow of the quantity per unit area an amount Fx ( xo x/, yo, zo) yz leaves the box at the front (Note that the area of the front face of the box is y z ) This quantity is the x-component of the flux multiplied by the area perpendicular to the x-direction x z y (x o, y o, z o ) y z x

22 The divergence of a vector We can also think of this quantity as x x Fx xo, yo, zoyz Fxo, yo, zoayz where A yz is a vector z whose magnitude is the area of the front surface of the box and y whose direction is outward x from the box The amount arriving into the box on the back face is similarly Fx ( xo x/, yo, zo) yz, (x o, y o, z o ) y z x

23 The divergence of a vector Hence the net amount leaving the box through the front or back faces is x x Fxxo, yo, zo yzfxxo, yo, zoyz x x Fxxo, yo, zofxxo, yo, zo xyz x Fx x y z x where we are assuming a very small box

24 The divergence of a vector We can repeat this analysis for each of the other two pairs of faces so, adding three such equations we can write for the total amount of flow leaving the small box per unit volume of the box i.e., dividing by V xyz F F x y Fz divff x y z

25 Particle current When we are thinking of flow of particles to conserve particles s. j p t where s is the particle density and j p is the particle current density The minus sign is because the divergence of the flow or current is the net amount leaving the volume (Note: this is particle not electrical current)

26 Particle current and the wavefunction In our quantum mechanical case the particle density is r,t so we are looking for a relation of the form s. j p t but with r,t instead of s To do this requires a little algebra and a clever substitution

27 Particle current and the wavefunction We know that which is simply Schrödinger s equation We can also take the complex conjugate of both sides Noting that then we have r, t 1 H ˆ r, t t i r, t 1 H ˆ r, t t i t t t i Hˆ Hˆ 0 t

28 Particle current and the wavefunction Presuming the potential V is real and does not depend in time and taking our Hamiltonian to be of the form ˆ H Vr m then ˆ ˆ HH VV m m

29 Particle current and the wavefunction So our equation i ˆ ˆ HH 0 t becomes i 0 t m Now we use the following algebraic trick

30 Particle current and the wavefunction Hence we have i t m s which is an equation in the same form as t with r,t instead of s as desired and i j p m So we can calculate particle currents from the wavefunction when the potential does not depend on time. j p

31 Particle current and stationary states This expression applies also for an energy eigenstate Suppose we are in the nth energy eigenstate En nr, texp i t nr Then i jpn r n r n r n r n r m, t, t, t, t, t

32 Particle current and stationary states In i j r r r r r, t, t, t, t, t pn n n n n m the gradient has no effect on the time factor so the time factors in each term can be factored to the front of the expression and multiply to unity i E E jpn r n r n r n r n r m i n r n r n r n r m n n, t exp i t exp i t

33 Particle current and stationary states In i pn, t n n n n m j r r r r r nothing on the right depends on time so the particle current j pn does not depend on time That is, for any energy eigenstate n jpn r, t jpn r Therefore particle current is constant in any energy eigenstate For real spatial eigenfunctions particle current is actually zero

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