Realistic Experiments for Measuring the Wave Function of a Single Particle
We suggest scattering experiments which implement the concept of
“protective measurements” allowing the measurement of the complete wave function even when only one quantum system (rather than an ensemble) is available. Such scattering experiments require massive, slow, projectiles with kinetic energies lower than the first excitation of the system in question. The results of such experiments can have a (probabilistic) distribution (as is the case when the Born approximation for the scattering is valid) or be deterministic (in a semi-classical limit).
Since the very early days of quantum mechanics, it is generally believed that the wave function of a single quantum system, say, a hydrogen atom in its ground state, is not a directly measurable quantity. Rather dV is simply the probability that the electron will be found within a volume element dV around the point . Since localization of the electron does destroy the wave function, we need an ensemble of many identical quantum systems - say many hydrogen atoms in the ground state - to measure the wave function. The latter is therefore believed to be only a mathematical symbol devoid of direct physical reality at the one-system (atom) level. Recently, it has been pointed out, however, that the standard framework of quantum mechanics does allow measuring the wave function of a single particle in a bound, stationary, state. The “protective” measurements required for this purpose involve weak, adiabatic interactions of the particle with the measuring device. This suggestions was discussed quite extensively in the literature. Refs. 4-6 contain some critical comments which were answered in Ref.7. Refs. 8-10 address peculiar features of the proposed novel measurements. Recent further elaboration of the basic idea in the context of the two-state vector formalism were presented in Refs.12,13.
In the protective measurements the “pointer” indicates not an eigenvalue, , of the measured operator , as is in the case in the standard ‘impulsive’ measurements, but rather the expectation value:
of in the state of interest.
In the case of impulsive measurements the initial wave function is reduced (“collapsed”) into one of the eigenstates of the operator , specifically to the state whose eigenvalue is indicated by the pointer. The probability of a specific outcome, , is:
Thus if is the position operator the probability of finding particle at is
with the wave function in the usual coordinate basis.
The recently suggested protective measurements are radically different:
(a) First, after the protective measurement of the operator is completed, the particle stays in the same initial state with a probability that can be made arbitrarily close to 1. Specifically, the excitation to other states, assuming these are at a minimal (energy) distance away, can be made exponentially small
with the duration of the adiabatically switched interaction.
(b) Second, the result of the measurement, i. e. the pointer position is deterministic (rather than probabilistic) and equals the average of in the state . The accuracy of this last statement depends on the validity of first order perturbation in the interaction coupling the particle and measuring device, i.e. on the assumption that during the measurement process the wave function is only weakly distorted.
Using feature (a) we can repeat the protective measurement for N different operators . Knowledge of allows inferring the initial wave function with precision . We also need impulsive, position, measurements in order to sample the probability at points , and get to a requisite precision. There is, however, one crucial difference: Our original wave function is not destroyed by the repeated protective measurements so that one bound electron is all that is needed.
The purpose of the present paper is two-fold. First, we would like to point out (almost) “real experiments” in which we can, by repeated scattering on the same quantum system, say hydrogen in the ground state, map out the complete wave function, without ever ionizing or exciting the system. This is facilitated simply by using massive slow projectiles such that on the one hand their momentum is large enough to allow probing the atomic wave-function of interest and on the other their kinetic energy is smaller than the energy gap between the ground state and the first excited state. Our second result is that the two aspects (a) and (b) of the recently suggested protective measurements-namely the non-destruction of the wave function and the deterministic rather than probabilistic result of the measurements - are really independent. Thus we could have scattering experiments involving heavy projectiles which can never destroy the state of interest, i. e. which are completely elastic, where the elastic scattering is probabilistic. This in particular is the case when the Born approximation is applicable. The result of individual scattering could also be deterministic - e.g., when an “eikonal approximation” involving the exchange of many quanta is applicable.
The plan of the paper is the following. In Section II we briefly review the protective measurements of Aharonov, Anandan, and Vaidman (Ref. 3). Section III recalls the general connection between measurements and scattering experiments. It also explains why generically such experiments are of the “impulsive” type and tend to break the bound state. Section IV contains our main result namely the description of scattering experiments involving heavy projectiles so that they do not excite the target yet map out the wave function. For concreteness we assume that the scattering can be described in Born approximation. Section V extends the discussion to a case where many quanta are exchanged in typical scattering. The resultant deflection is then (for a given initial impact parameter ) practically deterministic. This corresponds more closely to the originally suggested scheme of Aharonov and Vaidman and Aharonov, Anandan and Vaidman.
Finally, in the summary in Section VI, we speculate on real experiments and discuss various putative heavy projectiles, which can implement these experiments.
II. The General von Neumann Formalism and the Novel Protective Measurements.
Let us first briefly review the basic von Neumann formalism underlying all quantum measurements, impulsive and protective, alike. The system of a particle of mass , coordinate , momentum and the measurement apparatus (“pointer”) with coordinate , momentum and mass are initially non-interacting. The Hamiltonian then separates into a sum of two parts pertaining to the particle and pointer respectively:
with , binding potentials. The corresponding initial state factorizes into two parts. The particle is assumed to be initially in a state which is a stationary bound state, say the ground state, in the potential . The pointer is assumed to be initially in some wave packet state, say a Gaussian: centered around the origin in coordinate space [or alternatively as the Fourier transformed Gaussian in momentum space, with ]. The measurement process of an operator consists of turning on, an interaction of the form:
between the system and pointer. The time profile of switching the interaction on and off is normalized by
In the impulsive measurements differs from zero only over a short time interval . During this time is very large and we can neglect in evaluating the evolution of the system
The initial state of the particle is decomposed in the basis of
. Upon applying each component acquires the appropriate phase:
where is the pointer wave function shifted via the translation operator from the initial position around , to a similar Gaussian packet centered around (so that the pointer indicates the specific eigenvalue measured).
Let us next consider the protective measurements suggest by AV and AAV. In these measurements is turned on for a long time . Eq. (7) implies then a very weak : . If the turning on and off of the interaction is sufficiently smooth then we can use
(i) the adiabatic approximation for treating the effect of . The weakness of allows us, furthermore, to do this
(ii) in a first order perturbation for .
Thus (i) implies that the original state gets continuously deformed to with being the instantaneous eigenfunction of the time-dependent Hamiltonian with eigenvalue :
and we have:
The probability of leaving the system in some excited final state is exponentially small:
The second feature implies (ii) that the energy shift can be evaluated by first order perturbation in the weak interaction :
The corresponding time evolution of the pointer will be given by
i. e. the Gaussian wave packet of the pointer will shift by
. Such a shift can be discerned if it is larger than the original width (in configuration space) of the wave packet, i. e. if
Using Eq. (12), the fact that the transition frequencies to all the higher levels exceed , and completeness:
it can be shown that the sum of all excitation probabilities to any bound (or unbound) excited state satisfies
Thus such a measurement indeed “protects” the initial state and maintains
it for subsequent experiments of the same protective nature, where other
, are measured.
In particular, AV and AAV choose the operators to be
, the projections onto many small space regions around
points . Since the ground state wave function is real and
fixes . Letting
recuperates the complete wave function.
III. Measurement as Scattering Experiments.
The formal von Neumann pointer-system interaction is similar to usual scattering. Let us scatter form the electron, bound to infinitely heavy “proton” , at the origin, a projectile particle, . The latter is initially moving with velocity along the axis, say. The scattered projectile particle plays here the role of the pointer. The interaction shifts its momentum by and deflects it by an angle . This deflection is finally translated, in the subsequent free evolution of the scattered wave packet, into a lateral displacement on a far-away screen.
A large class of , neutron or electron scattering, measure the “Form Factor” of quantum systems. The differential scattering cross section of electrons, say, in the Born approximation, is proportional to with the “Form Factor”:
and the charge density. If the charge density is contributed by one electron in the bound state, then is given by:
Once is measured, we can infer . For the case , is real and can also be determined. By (inverse) Fourier transformation, we would then find However, in general, this measurement cannot be done by scattering on one single atom and an ensemble of atoms is required. This is related to the dual significance of the form factor in impulsive scattering; namely that is then also the probability of not destroying the system in any, specific, scattering event. Indeed, for collision time short in comparison with atomic periods, the “sudden” approximation is appropriate. The transfer of momentum transforms the initial wave function via:
The probability of staying in the ground state is then the square of the form factor:
The normalization condition = 1 implies that . However,
The probability that the initial atom stays in the ground state after repeated scattering with momentum transfers , , is therefore exponentially small:
[To show this recall that the geometric mean of positive numbers is smaller than the arithmetic mean. Hence:
with an effective average of obtained by the
experimental sampling.] Consequently, we need atoms if we want
to measure the form factor at points.
IV. “Protective” Scattering: The Born Approximations
We have seen that typical elastic scattering experiments are similar to impulsive von Neumann measurements. We proceed next to discuss adiabatic scattering corresponding to protective measurements.
If is small in comparison with the initial momentum of the projectile, then the deflection angle is small:
The effective collision time, estimated via the classical traversal time, is then:
with the effective size of the state . By the uncertainty principle the average (rms) momentum of the electron satisfies
If is the electrons’ mass, then:
are the electron’s “velocity”, the energy difference to the first excited state, and the atomic period respectively. The sudden approximation used above was predicated on the assumption that
or, using Eqs. (30) and (26) on:
If, further, , this implies a kinetic energy of the projectile, , which considerable exceed and the atom can be readily excited or ionized we are however interested. In the opposite, adiabatic case,
In order to be able to effectively probe over the whole relevant range of , we need that:
Clearly, Eq. (36) is consistent with Eq. (34), providing that the projectile-electron mass ratio is large enough:
Let us impose next a stronger version of the adiabatic condition namely so that
In this case, the complete kinetic energy of the projectile is insufficient for exciting the atom. We then have only elastic scattering with the atom staying always in the ground state. We also take the mass of the nucleus to be so large that its recoil velocity and recoil energy are completely negligible.
For the rest of this section, we will assume that the projectile-electron interaction is weak enough so that the scattering cross section can be computed in the Born approximation:
with the form factor defined in Eq. (18) above and the interaction potential in Fourier space:
(Thus for Coulomb interaction . By measuring , we can then infer and . Since the atom is never excited, we need just one atom from which we repeatedly scatter (a beam of ) many projectiles. It is amusing to ask how much time is required in such an experiment in order to map out the ground state wave function to within an accuracy of 10. For an incident flux the rate of collision will be with the elastic cross section. (The parameter expresses the weakness of the interaction.) In order to obtain an precision in the determination of (or ), we need scattering. The “experiment” should therefore be carried over a time such that:
Two considerations limit the flux with n the number density of projectiles in the beam. First, from Eq. (38) . Also we take the number density of projectiles , to avoid simultaneous scattering of several projectiles. Thus, we find
with a measure of how far we are from the “maximal density” . Even for a large , a small (guaranteeing the validity of the Born approximation) and a tiny :
We need “only” in order to obtain
This is due to the very short atomic period, , which for hydrogen, is sec.
V. Protective Scattering: The Semi-classical Approximation.
In the previous section we have shown that scattering of slow massive projectiles, when treated in the Born approximation, can allow finding the full form factor and hence the complete wave function of the atom, even if only a single atom is available. While this demonstrates the measurability of the wave function of a single system, the scattering considered differ form the protective measurements of Aharonov and Vaidman and Aharonov, Anandan, and Vaidman in one crucial respect. The outcome of any single scattering was probabilistic with the probability for scattering with momentum transfer being proportional to -whereas in the AV and AAV protective measurements, the pointer deflection [i. e. scattering angle] was deterministic.
As will be shown next, the latter, deterministic case, can also be modeled by semi-classical scattering. To this end we will maintain the conditions for protective scattering. (i) The kinetic energy of the projectile satisfies so that inelastic scattering cannot happen, though . (ii) The projectile-electron interaction is weak enough so that the original wave function is not distorted. We will however relax the assumption that scattering can be treated in the Born approximation. For convenience we will assume a coulombic interaction between the electron of change q and projectile of change :
[Taking q and to have the same sign disallows rearrangement reactions where the electron is picked up by the projectile.] We really need not specify the binding potential . However, to make the discussion more concrete, we also take a Coulombic binding potential:
Our requirement is implemented by having
To simplify the treatment, we assume that there is no direct projectile-nucleus interaction. We assume as in (ii) above that constitutes a small perturbation for the electron. We will not assume that it is weak insofar as the projectile motion is concerned. The weaker projectile ( particle) – electron interaction as compared with electron-proton interaction is conveyed in Fig. 1a by the larger number of virtual quanta (photons exchanged) between and .
Figure 1: Scattering of a heavy projectile on the light electron bound strongly to a super heavy proton . The thickness of the lines conveys the information on the masses of the particles. (a) The scattering in a semi-classical, many photon exchange, approximation. (b) The scattering in the Born, one photon exchange, approximation.
The phase picked up by the projectile as it moves along a straight line parallel to the z axes at an impact parameter with fixed velocity is approximately given by:
with a cut-off length. By having
we make the projectile electron interaction strong. The large Coulomb phase and the large angular momentum
suggest that we can treat the projectiles’ motion semi-classically. The Feynman diagrammatic expression of this is the fact that the momentum transfer is built up by exchanging a large number of photons, see Fig. 1a, rather than via a single photon exchange as in the case of the Born approximation, see Fig 1b . A semi-classical computation of the deflection, assuming still, for convenience, that -the scattering angle is small, can be done as follows. The electrostatic force exerted by the (spherically symmetric!) charge distribution of the electron on the projectile is
where Q(R) is the total electric charge within a sphere of radius
[We note that the use of the average value (for the charge inside R) in the
state parallels the general discussion in Section II above where
determined the deflection of the pointer.] The momentum transfer is then:
where we used and the fact that the net momentum transfer in this approximation [with the projectile moving with uniform velocity parallel to the axes] is purely transverse i. e. along . The order of magnitude of is readily estimated:
Such a is indeed expected if, as conjectured before, we have photons exchanged, each contribution a typical momentum transfer of . This multiple exchange make the total momentum transfer larger by a factor and easier to measure. Also the large number of quanta involved tends to make the deflection almost deterministic so that the above classical treatment of the scattering is justified. [Parenthetically we note that a small deflection:
requires a stronger version of Eq. (36).
While the last equation implies that
the latter does not conflict with the basic requirement
proving that the ratio is large enough. This however is precisely our Eq. (49) above.]
To measure the wave function, it suffices to find Q(r) of Eq. (54) since is given by:
Scattering projectiles with various initial impact parameters b and measuring in each case the momentum transfer given via Eq. (55), we can solve for . [In practice also such classical scattering, where is uniquely determined for a given b by a calculable function , are carried out by using an incident uniform beam of projectiles and measuring .]
VI. Summary and Discussion.
The arguments of this paper, complementing and slightly extending the
original work of AAV and AV definitely show that there is no quantum
mechanics subtlety hinders the measurement of the wave function of a single
atom. In all cases we require a projectile which is much heavier than the
particle say electron whose wave function is being measured.
The set-ups considered were still somewhat idealized. Thus in Section V we
assumed that there is a nucleus-electron and electron-projectile
interaction, but no direct interaction between projectile and nucleus.
These interactions could be Coulomb-like only if the electron carried two
types of charges: , and
sharing with the nucleus one type of charge and with the projectile the
other type. While the absence of projectile nucleus interaction is not an
absolute must, it excludes the possibility of the projectile being captured
around the nucleus. Ideally, the projectile should also:
(a) be point like;
(b) have long lifetimes;
(c) have minimal interaction with the nucleus
Clearly (a) - (c) are a “convenience” issue. Thus in principle we could use whole atoms or ions as massive slow projectiles. The separation of the target wave function from the effects due to the structure of the projectile, would then be more difficult. The muon is point like and . However, to satisfy the basic condition (37):
we need very slow muons. The muons’ lifetime is only seconds and these slow muons propagate only a few centimeters before decaying. The task of generating intense, monochromatic, collimated muon beams, scattering them and later, detecting the scattered muons seems very difficult.
We could also utilize neutrons which are heavy and fairly long lived. However, the low energy neutrons often have strong nuclear interactions which could mask the effects of the electron-neutron scattering.
Amusingly, the scattering of hypothetical WIMPs (“Weakly Interacting Massive Particles”) of mass, say on heavy nuclei , provides a simple example of a form factor appearing when the interaction is purely elastic. For , the kinetic WIMP energy: (for , typical virial velocity) is below nuclear excitation. Thus our basic no excitation requirement (Eq. (38a)) is readily satisfied. At the same time the maximal momentum transfer is appreciable and so that could be significantly smaller than unity.
In conclusion, it is amusing to note that quantum systems are actually more rigid than classical systems. The latter have continuous spectra and can suffer arbitrarily small excitation. Any scattering from a classical target would leave the latter slightly deformed. This however is not the case for quantum mechanical system
I have benefited from hearing first-hand about the measurement of the wave function from the authors of the original papers (Y. Aharonov, J. Anandan, and L. Vaidman). The trigger for the present work, beside the desire to have more concrete realization of their ideas, was a comment by F. T. Avignone concerning the form factors for scattering of WIMPS for which I am particularly grateful. The author would like to acknowledge the USA – Israel Science Foundation.
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