Section 2: Wave Functions and Probability Solutions

Transcription

1 Phsics 43a: Quantum Mechanics I Section : Wave Functions and Probabilit Solutions Spring 5, Harvard Here is a summar of the most important points from the second week with a few of m own tidbits), relevant for either solving homework problems, or for our general education This material is covered in Chapter of [] The dnamics of a quantum particle of mass m is governed b a complex-valued wave function, Ψx, t), obeing the Schrödinger equation i Ψ t = Ψ + V x)ψ ) m x We interpret Ψ as the probabilit densit of where the particle is located, so that the probabilit that we measure a < x < b is Pa < x < b) = We will also normalize our wave function so that b a dx Ψx, t) ) dx Ψ = 3) The expectation values of the position/momenta of the particle is given b x = p = i dx x Ψ, dx Ψ Ψ x 4a) 4b) After we measure the position of the particle, the wave function collapses on to a wave function localized around x = x For now, think of Ψx, t) dx as the probabilit that we would measure the particle in between points x and x + dx at time t The Schrödinger equation and thus quantum mechanics) is deterministic Randomness comes into the stor from the wave function collapse picture above This collapse picture is satisfactor for understanding experiments toda, but almost certainl is not the full stor We believe that the entire universe is governed b quantum mechanics, but we do not have a good phsical understanding of measurement For commentar see []

2 Here are two problems to practice the basic phsics learned this week that will be relevant on our first problem set: the first problem is meant to be a primer in probabilit and is much more subtle than an statistics ou ll ever have to do for this course!), and the second will help ou get some intuition behind manipulations of the time dependent Schrödinger equation I also hope ou ll walk awa having learned something new and interesting Problem The Distribution of Incomes): In this problem, we will explore a simple model for how incomes are distributed within a societ, that turns out to match reasonabl well with empirical data [3] Let s suppose that we have an econom consisting of N people, who report their incomes Y i i =,, N) each ear to the IRS Then the IRS can go ahead and collect all this data and report back the distribution of incomes For theoretical purposes, let s suppose that the reported back a smooth function ρ), which is defined so that for most reasonable intervals, d ρ) = number of people with income Y j, such that < Y j < N We ll refer to the object on the left hand side of this equation as the probabilit that someone in our econom has an income in between and a) What is the probabilit that someone in our econom has income exactl equal to? Solution: The probabilit that we have an income of exactl is, since is a continuous variable and we have a finite number of people in the econom! More carefull, let s ask for the probabilit that we have income in between and + ɛ, and then take ɛ : P < < + ɛ) = +ɛ d ρ) ρ )ɛ + Oɛ ) Thus as ɛ, the probabilit of obtaining exactl indeed goes to Next, let us construct a ver simple model that matches quite well with empirical data on the incomes of the rich in basicall ever single countr The full model is rather annoing to solve exactl, so we ll break it down into two parts The basic idea is that the poor and the rich have two predominantl different mechanisms b which their incomes change from ear to ear So we ll first stud the poor, and then the rich Suppose that ou have income Y t) in ear t, and income Y t + ) in ear t + We re interested in Y = Y t + ) Y t) When ou are poor, changes in our income are predominantl random, and so ou re probabl equall likel to see our salar go up and go down in an average ear However, because of the effects of inflation, it turns out that ou re more likel to lose mone in real terms) in an given ear The simplest possible model that accounts for this is simpl to sa that P Y = α) = P Y = β) =, with α > β Half the time ou lose income α, and half the time ou gain income β Now, we want to construct ρ) for an econom where approximatel everbod is poor To do this, we can use the following logic Let us consider a fixed income We need to count the probabilit p + ) that an individual started out below income in ear t, and ended up above income in ear t + ; and p ), the probabilit that we started out above in ear t and ended up below in ear t + In order for the distribution ρ) to be stationar, we must have p + ) = p ) for ever

3 b) Assume that α and β are small enough so that ρ is ver smoothl varing on these income scales, and ma be treated as an approximatel linear function for the purposes of computing p ± ) Use the condition p + = p to find a differential equation for ρ) Solution: Let s begin b counting p + ) Since everone whose income goes up goes up b the amount β, then p + ) = d ρ ) ρ β ) β β ρ) β ) d β In the first equalit, we have integrated over the fraction /) of people whose incomes go up, and integrated over all initial incomes for which + β > The second inequalit is basicall a trapezoidal approximation to the integral In the third step, we use the fact that ρ is slowl varing on the income scale β We can do a similar argument and compute p ) = +α d ρ ) ρ + α ) α α ρ) + α ) d As noted previousl, if ρ) is stationar, then there can be no flux of probabilit through an, so p + ) = p ) This gives us the equation β α)ρ = α + β d c) Let us assume that nobod in our econom can have income < Solve the ODE for ρ), restrict to the interval < <, and find the normalized solution to the ODE You should find ρ) e λ What is λ in terms of α and β? Solution: If we define λ then our differential equation above becomes α β) α + β, We obtain ρ = d = λρ d λ = log ρ = λ + constant = ρ) = Ae λ with A an undetermined constant We can fix A b requiring that the probabilit distribution is normalized: = This fixes A = λ, so ρ) = λe λ d ρ) = d Ae λ = A λ 3

4 d) Among the poor, compute and σ, the mean and standard deviation of the income distribution Solution: We compute the mean using = d ρ) = d λe λ This can be done easil in Mathematica, or analticall though integration b parts, but here is a ver simple trick Note that d n e λ = d So from here we easil read off The standard deviation follows as well: ) n e λ = ) n d e λ = λ λ = = λ λ = λ σ = = d λe λ = λ, λ λ = λ ) n λ λ = n! λ n+ So we see that = σ = λ This basicall means that the poor tpicall have λ income, and fluctuations are definitel present, but are not too severe You ll probabl have an income of about = to = λ or so Next, let us consider the econom of the rich Incomes of the rich are often tied to investments and the stock market, and as such grow proportionall to the amount of income the currentl have: eg, if the stock market doubles, then our income doubles no matter how high it was to begin with So this time, we sa that P Y = a) = P Y = b) =, namel that half the time incomes grow b b, and half the time the fall b a e) Repeat the analses of parts b) and c), this time for the econom of the rich You should find that the solution to the resulting differential equation is What is γ in terms of a and b? Solution: We do exactl the same thing as before: p + ) = p ) = b +b ρ) γ d ρ ) b d ρ ) a ρ) b ρ) + a ), d ) d 4

5 This time we find the equation Define We can integrate our ODE to find a b) a + b ρ = d γ = a b) a + b ρ = log ρ = γ d = γ log + constant Thus we find ρ) = A γ for some undetermined constant A f) Realistic distributions for the incomes of the rich are alwas normalizable in the interval < < What are the constraints on γ? Solution: We need to normalize the distribution in the interval < < : A A = d ρ) = d γ = γ γ γ > γ Obviousl onl the case γ > is allowed, and we find the normalization constant A = γ ) γ g) Compute and σ Show that the ma be infinite Solution: We start with = d γ ) γ γ = γ γ Of course, this integral is onl convergent if γ > ; if γ <, =! Next we have γ = d γ ) γ = γ γ 3, σ = = γ γ ) γ 3 γ ) and σ are onl finite if γ > 3 These distributions can thus have infinitel large fluctuations or even infinitel large averages! This is often referred to as fat tails For these distributions, is the size of a tpical income and fluctuation so long as γ > 3, but if γ 3 is ver small, then there can be an anomalousl large prefactor governing the size of fluctuations there are some individuals who are ver rich! 5

6 h) Focus on the interesting case where < γ < If the econom has N r rich people, estimate the highest income among the rich, in a tpical realization of random incomes drawn from ρ) Compare to the result of part g), and comment Solution: When < γ <, we know that = Let s investigate further If we have N r rich people in the econom, we expect to find that number of rich with incomes > = N r d ρ) = N r d γ ) γ γ = ) γ N r As, the expected number of rich people that have more than income vanishes And in particular, we should onl expect a tpical econom to have a person of wealth if ) γ N r Thus we find that the richest person in the econom probabl has income = N /γ ) r Interestingl, the income of the richest person in the econom grows quite dramaticall) with the number of rich in the econom For this econom it does not make sense to think about the average wealth of a rich person, because this average is dominated b the richest person in the econom! For tpical econom N r + N γ)/γ ) r + The sole richest person in the econom contributes more to the net income of the econom than almost everone else combined! And this is basicall wh is divergent fluctuations in how rich the richest person is are huge Empirical income distributions tend to have γ Problem The Classical Limit): We know from everda experience that classical Newtonian mechanics F = ma) is a ver good wa to describe the motion of big, classical objects that we see around us It is not at all obvious how to recover this as an limit of the quantum Schrödinger equation, which we ve learned for a point particle of mass m in one dimension The basic intuition is that classical phsics occurs when However, we can t just go ahead and set = above, because that trivializes the entire equation! We have to tr a bit harder A helpful place to start is to re-write the wave function as where ρ and θ are real-valued fields Ψ = ρe iθ, a) Plug in the ansatz for Ψ into the Schrödinger equation Show that taking the real and imaginar parts of the resulting answer, and after performing a few more basic manipulations, ou find the pair of equations t ρ + x ρv) =, t v + v x v = m x V ) x ρ m ρ 6

7 where v m xθ Solution: Let us plug the ansatz Ψ = ρe iθ into the time-dependent Schrödinger equation: i tρ ρ ) ρ t θ e iθ = V ρe iθ m x ρ + i x ρ x θ + i ρ xθ ρ x θ) ) e iθ ρ and θ are real, so we can take the real and imaginar parts of this equation The imaginar part reads [ t ρ + x θ x ρ + ρ ρ m xθ )] e iθ =, and defining v as in the problem, The real part gives us Multipl b /m) x ; we find t θ = V t ρ + x ρv) = t v + v x v = m x ) x ρ x θ) m ρ V ) x ρ m ρ b) The equations above ma remind ou of the equations of motion of a compressible fluid, up to the remnant of in the second equation Explain the phsical content contained in each of these equations How have we managed to hide? Solution: The first equation encodes conservation of probabilit densit, as discussed in class The second equation is the analogue of momentum conservation Newton s Laws for a fluid up to the external forcing due to the external potential V The onl place explicitl appears is a densitdependent correction to the potential energ an analogue of pressure) The reason doesn t show up much anmore is that v is proportional to In realit, this means that it is the rapid fluctuations of the phase that carr much of the dependence of Ψ It now suffices to show that these fluid-like equations reduce to Newton s Law F = ma We still need to make a phsical assumption namel, that the quantum particle is approximatel localized at the point X, but is spread out over a small length scale l For example, ou might think ρ e x X /l / πl l is effectivel an input parameter, and there is no forced relationship between l and an other parameter in the problem, though later on in the course we ll see that l would tpicall be at least as big as some minimum l c) Suppose that a classical point particle at point X would feel a force F How large must l be in order for us to neglect the term in the equations of part a)? Solution: Note that x V F as in classical mechanics, the overall constant scale of V is not important for phsics) However, the fluctuations in densit are phsical, and so we must take x ρ ρ/l We conclude x V m F m m x ρ x ρ ml 3, We ma crudel estimate xρ ρ/l for the purposes of estimating whether terms are big or small 7

8 or ) /3 l mf From now on, we ll take l to be much larger than the required value found in part c) The classical limit will reall consist of first, and l second What we reall care about is that V is approximatel constant on the length scale l: namel, that V /l x V d) Justif defining the approximate position of our classical point mass to be Xt) dx xρx, t) Solution: Recall from the summar of the lectures that x = dx x Ψ = dx xρ When ρ is a sharpl peaked function, we conclude that it is reasonable to approximate the position of a classical particle, X x e) Manipulate the equations of part a), and show that under reasonable approximations m d X dt V x x=x This is, of course, another wa of saing that F = ma Solution: Let s begin b computing dx dt = dx x t ρ = dx x x ρv) = dxρv, where we have integrated b parts in the last step Next, we note that, within our approximations, We conclude m d X dt ρ m xv = v t ρ + x ρv)) + ρ t v + v v x) = t ρv) + x ρv ) = m dx t ρv) = m dx ρ m xv x ρv )) = dx ρ x V For the classical particle localized on scales where x V is approximatel constant, the integral on the right hand side above is approximatel x V )X) We conclude that m d X dt V x x=x On the homework, ou are supposed to prove an exact theorem of quantum mechanics that, using the same approximations we have made, reduces to a derivation of Newton s Laws You ma use ver similar logic to this problem if ou wish, but it will probabl be easier to work directl with Ψ and Ψ, and integrate b parts 8

9 [] D J Grififths Introduction to Quantum Mechanics Prentice Hall, nd ed, 4) [] S Weinberg Lectures on Quantum Mechanics Cambridge Universit Press, 3) [3] V M Yakovenko and J B Rosser Colloquium: Statistical mechanics of mone, wealth and income, Reviews of Modern Phsics ), arxiv:9558 9