What we did not learn in our class was that, even back at that time, there had been several clever experiments with neutrons which demonstrate the influence of the gravitational potential on the phase of the neutron wave function using interferometers. Neutrons, of course, are ideal particles to perform such experiments, since they have no electric charge and are not subject to the influence of the ubiquitous electromagnetic fields.
But only over the last few years, new experiments have been realised that show directly the quantisation of the vertical "free fall" motion of neutrons in the gravitational field of the Earth. I had heard about them some time ago in connection with their possible role for the detection of Non-Newtonian forces, or the modifications of Newtonian gravity at short distances. Then, earlier this year, I heard a talk by one of the experimenters at Frankfurt University, and I was quote fascinated when I followed the papers describing the experiments.
The essential point of these experiments is the following: If you prepare a beam of very slow neutrons - with velocities about 10 m/s - you can make them hop over a reflecting plane much like you can let hop a pebble over the surface of a lake. Then, you can observe that the vertical part of the motion of the neutrons - with velocities smaller than 5 cm/s - is quantised. In fact, one can detect the quantum states of neutrons in the gravitational field of the Earth! Let me explain in more detail...
Free Fall in Classical Mechanics ...
In order to better understand the experiment, let's go back one step and consider the very simple motion of an elastic ball which is dropped on the ground. If the ground is plane and reflecting, and the ball is ideally elastic such that there is no dissipation of energy, the ball will jump back to the height of where it was dropped from, fall down again, jump back, fall, and so on. The height of the ball over ground as a function of time is shown as the blue curve in the left of this figure: it is simply a sequence of nice parabolas.
We can now ask, What is the probability to find the bouncing ball in a certain height above the floor? For example, we could make a movie of the bouncing ball, take a still at some random time, and check the distribution of the height of the ball if we repeat this for many random stills. The result of this random sampling of the bouncing motion of the ball is the probability distribution shown in red on the right-hand side of figure. The probability to find the ball at a certain height in this idealised, "stationary" situation, where the elastic ball is bouncing forever, is highest at the upper turning point of the motion, and lowest at the bottom, where the ball is reflected.
... and in Quantum Mechanics
So much for classical mechanics, as we know it from every-day life. In quantum mechanics, unfortunately, there is not anymore such a thing as the path of a particle, with position and velocity as well-defined quantities at any instant in time. However, it still makes sense to speak of stationary states, and of the probability distribution to find a particle at a certain position. In quantum mechanics, it is the wave function which provides us with this probability distribution by calculating its square. And the law of nature determining the wave function is encoded in the famous Schrödinger equation. The Schrödinger equation for a stationary state is an "eigenvalue equation", whose solution yields, at the same time, the wave function and the value of the energy of the corresponding state. For the motion of a particle in a linear potential - such as the potential energy mgx of a particle with mass m at height x above ground in the gravitational field with acceleration g at the surface of the Earth - it reads
In some cases, there are so-called "exact solutions" to the Schrödinger equation - wave functions that are given by certain functions one can look up in thick compendia, or at MathWorld. These functions usually are some beautiful beasts out of the zoo of so-called "special functions". Such is the case for the motion of a particle in a linear potential, where the solution of the Schrödinger equation is given by the Airy function Ai(x). Interestingly, this function first showed up in physics when the British astronomer George Airy applied the wave theory of light to the phenomenon of the rainbow...
Quantum States of Particles in the Gravitational Field
As a result of solving the Schrödinger equation, there is a stationary state with a minimal energy - the ground state - and a series of excited states with higher energies. Here is how the wave function of the second excited state of a particle in the gravitational field looks like as a function of the height above ground:
The wave function, shown on the left in magenta, oscillates through two nodes, and goes down to zero exponentially above the classical limiting height, which corresponds to the upper turning point of the parabola of a classical particle with the same energy. For neutrons in this state, this height is 32.4 µm above the plane. The green curve on the right shows the probability density corresponding to the wave function. It is quite different from the classical probability density, shown in red. As a characteristical property of a quantum system, there is, besides the two nodes, a certain probability to find the particle above the classical turning point. This is an example of the tunnel effect: there is a chance to find a quantum particle in regions where by the laws of classical physics, it would not be allowed to be because of unsufficient energy.
However, going from the ground state to ever higher excited states eventually reproduces the probability distribtion of classical physics. This is what is called the correspondence principle, and you can see what it means if you have a look at the wave function for 60th excited state: Here, the probability distribution derived from the quantum wave function follows already very closely the classical distribution.
So far, we have been talking about theory: the Schrödinger equation and its solutions in guise of the Airy function. There is no reason at all to doubt the validity of the Schrödinger equation: it has been thoroughly tested in innumerable situations, from the hydrogen atom to solid state physics. However, in all these situations, the interaction of the particles involved is electromagnetic, and not by gravitation. For this reason, it is extremely interesting to think about ways to check the solution of the Schrödinger equation for particles in the gravitational field. As we have seen before, the best way to do this is to work with neutrons, in order to avoid spurious electromagnetic effects.
Bouncing Neutrons in the Gravitational Field
Unfortunately, it is so far not possible to scan directly the probability distribution of neutrons in the gravitational field. However, in a clever experimental setup, one can look at the transmission of neutrons through a channel between with a horizontal reflecting surface where they can bounce like pebbles over a lake, and an absorber ahead. This is a rough sketch of the setup:
The decisive idea of the experiment is to vary the height of the absorber above the reflecting plane, and to monitor the transmission of neutrons as a function of this height. If the height of the absorber is to low, the ground state for the vertical motion of the neutrons does not fit into the channel, and no neutrons will pass the channel. Transmission sets in once the height of the channel is sufficiently large to accommodate the ground state wave function of the vertical motion of the neutron. Moreover, whenever with increasing height of the channel, one more of the excited wave function fits in, the transmission should increase. The first of these steps, and the corresponding wave functions and probability densities, are shown in this figure:
The interesting point now is, can this stepwise increase of transmission be observed in actual experimental data? Here are measured data, and indeed - the first step is clearly visible, and the second and third step can be identified:
Adapted by permission from Macmillan Publishers Ltd: Nature (doi:10.1038/415297a), copyright 2002.
This has been the first verification of quantised states of particles in the gravitational field!
What can be learned
You may wonder if the experiment may not have shown just some "particle in a box" quantisation, since the channel for the neutrons formed by the reflecting plane and the absorber may make up such a box. This objection has been raised, indeed, in a comment paper, and has been answered by detailed calculations, and improved experiments: the conclusion about quantisation in the gravitational field remains fully valid!
However, limits about modifications of Newtonian gravity from this experiment remain restricted. Such a modification would change the potential the neutrons are moving in. For example, a short-range force caused by the matter of reflecting plane could contribute to the potential of the neutrons. However, as comes out, such an additional potential would be very weak and have nearly no influence at all on the overall wave function of the neutron.
Moreover, it is clear that in this experiment, the gravitational field is always a classical background field, which itself is not quantised at all. There may be the possibility that a neutron undergoes a transition from, say, the second to the first quantised state, thereby emitting a graviton - similar to the electron in an atom, which emits a photon when the electron makes a transition. Unfortunately, this probability is so low that it is not reasonable to expect that it may ever be measured....
But all these restrictions do not change at all the main point that this a very exciting, elementary experiment, which could find its way into textbooks of quantum mechanics!
Here are some papers about the "bouncing neutron" experiment:
Quantum states of neutrons in the Earth's gravitational field by V.V. Nesvizhevsky, H.G. Boerner, A.K. Petoukhov, H. Abele, S. Baessler, F. Ruess, Th. Stoeferle, A. Westphal, A.M. Gagarski, G.A. Petrov, and A.V. Strelkov; Nature 415 (2002) 297-299 (doi: 10.1038/415297a) - The first description of the result.
Measurement of quantum states of neutrons in the Earth's gravitational field by V.V. Nesvizhevsky, H.G. Boerner, A.M. Gagarsky, A.K. Petoukhov, G.A. Petrov, H.Abele, S. Baessler, G. Divkovic, F.J. Ruess, Th. Stoeferle, A. Westphal, A.V. Strelkov, K.V. Protasov, A.Yu. Voronin; Phys.Rev. D 67 (2003) 102002 (doi: 10.1103/PhysRevD.67.102002 | arXiv: hep-ph/0306198v1) - A more detailed description of the experimental setup and the first results.
Study of the neutron quantum states in the gravity field by V.V. Nesvizhevsky, A.K. Petukhov, H.G. Boerner, T.A. Baranova, A.M. Gagarski, G.A. Petrov, K.V. Protasov, A.Yu. Voronin, S. Baessler, H. Abele, A. Westphal, L. Lucovac; Eur.Phys.J. C 40 (2005) 479-491 (doi: 10.1140/epjc/s2005-02135-y | arXiv: hep-ph/0502081v2) - Another more detailed discussion of the experimental setup, possible sources of error, and the first results.
Quantum motion of a neutron in a wave-guide in the gravitational field by A.Yu. Voronin, H. Abele, S. Baessler, V.V. Nesvizhevsky, A.K. Petukhov, K.V. Protasov, A. Westphal; Phys.Rev. D 73 (2006) 044029 (doi: 10.1103/PhysRevD.73.044029 | arXiv: quant-ph/0512129v2) - A long and detailed discussion of point such as the "particle in the box" ambiguity and the role of the absorber.
Constrains on non-Newtonian gravity from the experiment on neutron quantum states in the Earth's gravitational field by V.V. Nesvizhevsky, K.V. Protasov; Class.Quant.Grav. 21 (2004) 4557-4566 (doi: 10.1088/0264-9381/21/19/005 | arXiv: hep-ph/0401179v1) - As the title says: a discussion of the constraints for Non-Newtonian forces.
Spontaneous emission of graviton by a quantum bouncer by G. Pignol, K.V. Protasov, V.V. Nesvizhevsky; Class.Quant.Grav. 24 (2007) 2439-2441 (doi: 10.1088/0264-9381/24/9/N02 | arXiv: quant-ph/07702256v1) - As the title suggests: the estimate for the emission of a graviton from the neutron in the gravitational field.
TAG: physics, quantum mechanics