USU PHYS 3750 - The Uncertainty Principle

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Lecture 29 Phys 3750 D M Riffe -1- 4/10/2013 The Uncertainty Principle Overview and Motivation: Today we discuss our last topic concerning the Schrödinger equation, the uncertainty principle of Heisenberg. To study this topic we use the previously introduced, general wave function for a freely moving particle. As we shall see, the uncertainty principle is intimately related to properties of the Fourier transform. Key Mathematics: The Fourier transform, the Dirac delta function, Gaussian integrals, variance and standard deviation, quantum mechanical expectation values, and the wave function for a free particle all contribute to the topic of this lecture. I. A Gaussian Function and its Fourier Transform As we have discussed a number of times, a function ()xf and its Fourier transform ()kfˆ are related by the two equations () ()∫∞∞−= dkekfxfikxˆ21π, (1a) () ()∫∞∞−−= dxexfkfikxπ21ˆ, (1b) We have also mentioned that if ()xf is a Gaussian function ()22xxexfσ−= , (2) then its Fourier transform ()kfˆ is also a Gaussian, ()222ˆkkkekfσσ−= , (3) where the width parameter kσ of this second Gaussian function is equal to xσ2 , and so we have the result that the product kxσσof the two width parameters is a constant, 2=kxσσ. (4) Thus, if we increase the width of one function, either ()xf or ()kfˆ, the width of the other must decrease, and vice versa.Lecture 29 Phys 3750 D M Riffe -2- 4/10/2013 Now this result wouldn't be so interesting except that it is a general relationship between any function and it Fourier transform: as the width of one of the functions is increased, the width of the other must decrease (and vice versa). Furthermore, as we shall see below, this result is intimately related to Heisenberg's uncertainty principle of quantum mechanics. II. The Uncertainty Principle The uncertainty principle is often written as 2h≥∆∆xpx , (5) where x∆ is the uncertainty in the x coordinate of the particle, xp∆ is the uncertainty in the x component of momentum of the particle, and π2h=h, where h is Planck's constant. Equation (5) is a statement about any state ()tx,ψ of a particle described by the Schrödinger equation.1 While there are plenty of qualitative arguments concerning the uncertainty principle, today we will take a rather mathematical approach to understanding Eq. (5). The two uncertainties x∆ and xp∆ are technically the standard deviations associated with the quantities x and xp , respectively. Each uncertainty is the square root of the associated variance, either ()2x∆ or ()2xp∆, which are defined as ()()22xxx −=∆ , (6a) ()()22xxxppp −=∆ , (6b) where the brackets indicate the average of whatever is inside them. Experimentally, the quantities in Eq. (6) are determined as follows. We first measure the position x of a particle that has been prepared in a certain state ()tx,ψ. We must then prepare an identical particle in exactly the same state ()tx,ψ (with time suitably shifted) and repeat the position measurement exactly, some number of times. We would then have a set of measured position values. From this set we then calculate the average position x. For each measurement x we also calculate the quantity ()2xx −, and then find the average of this quantity. This last calculated quantity is the 1 We are implicitly thinking about the 1D Schrödinger equation; thus there is only one spatial variable.Lecture 29 Phys 3750 D M Riffe -3- 4/10/2013 variance in Eq. (6a). Finally, the square root of the variance is the standard deviation x∆ . This whole process is then repeated, except this time a series of momentum measurements is made, allowing one to find xp∆. What we want to do here, however, is use the theory of quantum mechanics to calculate the variances in Eq. (6). How do we calculate the average value of a measurable quantity in quantum mechanics? Generally, we calculate the expectation value of the operator associated with that quantity. For example, let's say we are interested in the (average) value of the quantity O for a particle in the state ()tx,ψ. We then calculate the expectation value of the associated operator Oˆ, which is defined as2 () ()()()∫∫∞∞−∞∞−=dxtxtxdxtxOtxO,,,ˆ,**ψψψψ. (7) The quantity O can be any measurable quantity associated with the state: the position x, for example. Notice that the variance involves two expectation values. Again consider the position. We see that we must first use Eq. (7) to calculate x and then use that in the calculation of the second expectation value. Finally to get x∆ we must take the square root of Eq (6a). We can actually rewrite Eq. (6) in slightly simpler form, as follows. Consider Eq. (6a). We can rewrite it as ()2222 xxxxx +−=∆ (8) Now because the expectation value is a linear operation [see Eq. 7], this simplifies to ()2222 xxxxx +−=∆ . (9) 2 Usually in quantum mechanics one deals with normalized wave functions, in which case the denominator of Eq. (6) is equal to 1. Rather than explicitly deal with normalized functions, we will use Eq. (7) as written.Lecture 29 Phys 3750 D M Riffe -4- 4/10/2013 Furthermore, because an expectation value is simply a number, 22xx = and 2222 xxxxx == . Eq. (9) thus simplifies to ()222xxx −=∆. (10a) Similarly, for ()2xp∆ we also have ()222xxxppp −=∆. (10b) Thus we can write the two uncertainties as 22xxx −=∆ , (11a) 22xxxppp −=∆ . (11b) III. The Uncertainty Principle for a Free Particle A. A Free Particle State Let's now consider a free particle and calculate these two uncertainties using Eq. (11). You should recall that we can write any free-particle state as a linear combination of normal-mode traveling wave solutions as () ()()[]∫∞∞−−=tkxkiekCdktxωπψ21,, (12) where the coefficient ()kC of the k th state is the Fourier transform of the initial condition ()0,xψ, () ()∫∞∞−−=xikexdxkC 0,21ψπ, (13) and the dispersion relation is, of course, given by ()mkk 22h=ω. To keep things simple, let's assume that the state we are interested in is a particle moving along the x axis. As discussed in the Lecture 27 notes one particular initial condition (but certainly not the only one, see Exercise 29.1) that can describe such a particle isLecture 29 Phys 3750 D M Riffe -5- 4/10/2013 ()22000,xxxikeexσψψ−= .


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USU PHYS 3750 - The Uncertainty Principle

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