Notes on wavefunctions II: momentum wavefunctions

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1 Notes on wavefunctions II: momentum wavefunctions and uncertainty Te state of a particle at any time is described by a wavefunction ψ(x). Tese wavefunction must cange wit time, since we know tat particles can move. For a traveling particle, we migt not be able to say exactly were te particle is at any given time, but te region were we are most likely to find it canges wit time as it travels along. Eventually, we would like a general equation tat determines exactly ow wavefunctions cange wit time. To start, we ll focus on te simple case of te wavefunction for an electron wit some specific momentum p. Wavefunctions for momentum states Let s first ask te question: wat does te wavefunction for a traveling electron look like? Here, we can use some of our experimental observations. Electrons wit momentum p produce a pattern on te screen in a double slit experiment just like te pattern from a wave wit wavelengt λ = /p. For all oter types of waves (water waves, sound waves, classical ligt waves), suc a pattern is explained by saying tat before te slits we just ave a pure sinusoidal wave wit some wavelengt traveling towards te apparatus, and tat tis wave goes troug te slits and spreads out again, wit waves from te two slits recombining at te screen to produce constructive and deconstructive interference. Since we get te same result for electrons, it makes sense tat te matematical explanation for te pattern sould be te same: te traveling electrons wit momentum p sould be described by pure sinusoidal waves wit wavelengt λ = /p. Now tat we understand tat electron states in general are described by wavefunctions, we can make a more precise proposal: te wavefunction ψ(x) for a traveling electron wit momentum p sould look like a pure sinusoidal wave wit wavelengt /p. Tis proposal turns out to be exactly rigt. Aside: complex numbers By tinking about te poton picture of polarizer experiments, we were led to te idea of quantum superposition. An important feature of tis is tat if a and b are are two states of a pysical system (peraps wit definite 1

2 values for some pysical property suc as position) ten we can also ave a state α a + β b. Up until now, we ave been assuming tat α and β are real numbers. However, in order to describe te most general states, we need to allow α and β to be complex numbers. In te polarization case, letting α and β be complex allows us to describe potons wit circular and elliptical polarization, rater tan just linear polarization. Tis is discussed in detail in tutorial 10, wit additional background material in te Notes on complex numbers. Te bottom line for te present discussion is tat wavefunctions ψ(x) tat are real functions are just special cases tat don t describe te most general electron states. Generally, te ψ at a location x will be a complex number. We can write tis as ψ(x) = ψ R (x) + iψ I (x) were te real functions ψ R (x) and ψ I (x) are known as te real part and te imaginary part of te wavefunction respectively. Alternatively, we can write ψ(x) = A(x)e iθ(x). were A(x) and θ(x) are te magnitude and te pase of te wavefunction. To compute te probability density from a complex wavefunction, te rule is tat we take te magnitude squared of te complex number. Tus, we ave: P (x) = ψ(x) 2 = (ψ R (x)) 2 + (ψ I (x)) 2 = (A(x)) 2. Back to wavefunctions for traveling electrons. We concluded above tat te wavefunction for a traveling electron wit momentum p sould look like a wave wit wavelengt /p. Te simplest guess would be ψ(x) = cos( 2πp x) As you learned in tutorial 10, tis is te real part of a complex wave ψ(x) = e i 2πp x. 2

3 Since wavefunctions can be complex, we could actually ave eiter of tese as possible wavefunctions. It turns out tat only te second one is correct as te wavefunction for an electron wit momentum p. Wile it s probably not possible to definitively conclude tis from wat we know so far, we can at least appreciate tat tere is a pysical difference between ψ(x) and ψ(x): in te second case, te probability density is constant: ψ(x) 2 = cos 2 ( 2πp x) + sin2 ( 2πp x) = 1, wile in te first case, te probability density oscillates: ψ(x) 2 = cos 2 ( 2πp x). Also, in te first case, replacing p by p gives us te same function, wile in te second case, replacing p by p gives us someting different. Electrons wit momentum p are definitely different tan electrons wit momentum p, so ψ seems like a better coice. 1 In summary, we ave decided tat te wavefunction (at some particular time) for an electron wit momentum p takes te form of a complex wave ψ p (x) = e i 2πp x. Since an electron in suc a state as a definite momentum p, we call suc a state a MOMENTUM EIGENSTATE. Wavepackets You migt already be botered by te fact tat (as we saw above) ψ p (x) 2 = 1. Our only real constraint on possible wavefunctions so far is tat teir probability densities sould integrate to one, and tis function clearly violates tis. Te problem is tat ψ p (x) 2 is constant, so taking tis literally, te electron is equally likely to be found anywere in te universe. Tis doesn t sound very reasonable, especially since we started out trying to describe electrons inside our double slit experiment. 1 Matematically, we ave tat cos( 2πp x) = 1 2πp 2 (ei x + e x. So an electron wit wavefunction cos( 2πp x) is actually a quantum superposition of states wit momentum p and momentum p. i 2πp 3

4 Te resolution of tis puzzle isn t really anyting complicated to do wit wavefunctions or quantum mecanics. It is simply tat real waves in nature are never actually described by perfect sinusoidal functions like cos(kx). Pysical waves always ave finite extent, like tis (just te real part is sown): Here, tere is still a clear wavelengt, but te wave doesn t go on forever. A wavefunction like tis can clearly be normalized so tat te probability density integrates to one. For a wavepacket, te wavelengt λ and te extend of te packet, wic we ll call x are two independent quantities. Tere is one obvious restriction: tat te extent of te wavepacket must be at least as big as a wavelengt. Tis gives us our first int of te HEISENBERG UNCERTAINTY PRINCI- PLE: for a wave wit a well-defined wavelengt (or equivalently momentum), te wavefunction must be nonzero over a range of values as least as large as tis wavelengt. Tus, knowing someting definite about te momentum of a particle restricts ow certain we can be about its position. Wavepackets as sums of pure waves. We now come to a matematical fact tat is crucial for our understanding of wavefunctions in quantum mecanics. We ave argued tat pysical traveling electrons sould be described by wavepackets rater tan pure (infinitely extended) waves. But it is a fact tat any suc wavepacket in fact any wavefunction at all can be written as a sum (i.e. a superposition) of pure waves wit various wavelengts. 2 For different wavefunctions, te amount of eac pure wave will be different, so we need to introduce a function A(p) tat tells us ow muc of te pure wave wit momentum p we ave (i.e. wit wat amplitude do we add te wave wit momentum p into our superposition). In terms of A(p) our claim is tat for any wavefunction ψ(x), we can write ψ(x) = p A(p)ψ p (x). (1) 2 Tis is covered in greater detail in te text, capter 40. Te simulation described in tutorial 11 is also very elpful for understanding tis. 4

5 Te pysical interpretation of tis expression is tat a general state can be tougt of as a quantum superposition of momentum eigenstates. Tis means tat generally states do not ave a definite value for momentum, but if we measure te momentum, te probability density for finding te value p is A(p) 2. So A(p) acts just like a wavefunction for momentum. Te formula (??) isn t quite rigt, since tere is actually a continuous range of possible momenta. Tus, we sould use an integral instead of a sum. If we use te explicit form of ψ p (x) given above, tis becomes ψ(x) = 1 A(p)e i 2πp x dp. (2) Te factor of in front is just a convention; we could ave defined A in a different way to include tis factor also. How can we find te momentum wavefunction A(p) if we know te wavefunction ψ(x)? It turns out tere is a nifty formula (tanks, matematicians!): A(p) = 1 2πp i ψ(x)e x dx. (3) Tis is almost te same as (??) except for te minus sign in te exponential. Going from ψ(x) to A(p) in tis way (and vice-versa) is known as a Fourier transform. For us, te most important point is tat tese formulae exist: given any wavefunction, we can in principle figure out wat superposition of momentum eigenstates tat wavefunction corresponds to. Once we know tis, we can figure out te probabilities for finding different momenta if we measure te momentum. As for position, immediately repeated measurements of momentum sould give te same result. Tus, after a measurement of te momentum, te state of te particle will be approximately a momentum eigenstate (i.e. we will ave a very spread out wavepacket wit a sarply defined wavelengt), since only in a momentum eigenstate can we be guaranteed tat te next measurement of momentum will ave a definite value (equal to te previously measured value). Since measuring momentum results in a spread-out wavefunction, it tends to decrease out knowledge about te location of te particle. Tis is anoter int of te Heisenberg Uncertainty Principle, wic we will now make more precise. 5

6 Te Heisenberg Uncertainty Principle We can now be more precise about te connection between position uncertainty and momentum uncertainty. A property of Fourier transformations is tat in order to describe a position wavefunction tat is sarply peaked (e.g. a very sort wavepacket), we must add up pure waves wit a very broad range of momenta. Conversely, if we combine pure waves wit a very narrow range of momenta, te resulting wavefunction will necessarily be very spread out. Quantitatively, we can prove tat x p 4π, (4) were x represents te uncertainty in position, i.e. te range of values of x over wic tere is a significant cange to find te particle, and p represents te uncertainty in momentum, i.e. te range of values of p over wic we are likely to find te momentum. Te precise definitions of x and p are discussed in te notes on expectation values and uncertainties, but a simple way to define tem is tat if we tink of ψ(x) 2 or A(p) 2 as a istogram for te possible values of position/momenta, ten x and p are te standard deviation. Equation (??) is te precise form of te Heisenberg Uncertainty Principle. Tis result as a dramatic pysical interpretation: it says tat it is impossible to know bot te position and te momentum of a particle at te same time. Te more accurately we know te position, te larger te uncertainty in momentum muc be. Similarly, te more accurately we know te momentum, te less certain we can be about position. Te uncertainty relation (??) as an inequality sign rater tan an equality since for some wavefunctions, bot position and momentum are very uncertain (can you tink of an example?). In tree dimensions, tere are similar uncertainty relations tat apply to te y and z directions, so te complete set of relations is: x p x 4π y p y 4π z p z 4π. (5) Here, te tree position uncertainties are all independent quantities. For example, if we ave a wavefunction tat is very spread out in x but localized in y, tat would correspond to a case wit large x and small y. 6