Quantum mechanics
The most important discovery in science was quantum theory and its consequence, or even better, its inevitable property, a property of reality: the Heisenberg uncertainty principle.
I have written previously on problems in quantum physics, especially on simultaneous measurement of velocity and position of a particle and why it is limited by precision. That imprecision is impossible to remove from the physics laws that govern the universe.
“The electron is not some kind of a point particle, it is more like a wave of probability in the quantum field, constantly interacting with other particles and fields, including the Higgs field. It is why it is not possible to measure its position with infinite precision.”
Not only are the science experiments limited by uncertainty, properties of elementary particles are also necessary to be limited by uncertainty for the universe to work properly. The elementary particle can not be contained into a infinitely dense point as the charge and then energy density would be also infinite. And no computer could write the value of position to infinite number of decimal places — which is required by an infinite precision measurement.
“Knowing without probability, with complete certainty what would happen to every particle would be impossible, it would require complete determinism, measurement with infinite precision and such requirements would result in even worse paradoxes then the measurement problem. “
The problem is whether quantum physics provides a full description of reality because uncertainty of a measured particle depends on how precisely the position had been measured.
“The uncertainty in the measurement is defined by the Planck constant, which is, fortunately for our reality and existence, very small, and has a very small effect on macroscopic objects. The fact there is uncertainty in simultaneously measuring position and velocity of a particle is a consequence of that: measuring first position and then velocity, is not the same thing as first measuring velocity and then position.
By measuring the position you use a photon with a certain wavelength, and then the precision of the position is determined by the wavelength of the same photon. Then, there is uncertainty in position of the particle because of the finite wavelength of the photon, that ultimately changes the velocity of particle because every photon has energy and transfers it to the particle. That uncertain energy transfer to the particle causes random change in velocity, which is the cause of uncertainty. The immediate subsequent measurement of the velocity will not be the same as if the position had not been measured to begin with.”
The microscope works the same way, the photon with a certain wavelength is emitted and reflects on the observed object and photon is directed toward the microscope aperture where it is observed by a scientist. For a regular microscope the observed object is made from many elementary particles, so quantum properties are not present or observed in the measurement. For the Heisenberg microscope, it measures only one elementary particle.
It is impossible to measure a particle without using a photon that contains information about the properties of the particle. Properties of particles in the universe are conveyed by gauge bosons of fundamental forces: for gravity it is the graviton, for electromagnetic force it is the photon, for weak force it is W and Z bosons, for strong force it is the gluon. Gauge bosons are mediators of fundamental forces and their purpose is to gauge properties of elementary particles and reality.
When a physics experiment is inconceivable to perform in practice (or impossible with recent technology) the experiment can be imagined and studied as a mind experiment. Which is easier to implement and can be repeated and better designed with every instance, until a better understanding of laws of science are determined. Heisenberg microscope is a mind experiment designed to reveal true properties of reality and quantum physics.
So how does the Heisenberg microscope experiment work?
The best possible microscope
Heisenberg microscope is special microscope that observes only one particle, for simplicity let it be an electron. Photon has quantized energy E=hf with a certain frequency f defined by Planck constant h. The photon with energy
is reflected on the observed particle, which can be observed to determine information about the position of a particle.
Naively, if there were no quantum physics, the Planck constant would be zero (h=0), and from the expression for the energy of the photon, every photon would have zero energy.
The photon can also interact with the electron and change its energy, after which another photon is emitted that can be observed.
Optical microscopes are limited by photon wavelength in nanometres, and that is why electron microscopes are a better option because the wavelength of the electron
with a large momentum is more precise then optical light, (the electron can have a smaller wavelength as its energy can be larger then the energy of the photon in optical spectrum).
Velocity of the electron wavefunction
Momentum p (the same is for velocity p=mv) is defined mathematically by derivative with respect to position of particle r:
“Uncertainty is also a consequence from the fact that, as known from Newton and Leibniz and development of calculus, velocity is a derivative and the rate of change with position and time. That is why velocity and position have a special connection, and are connected by uncertainty, because change in position requires change of velocity, and change of velocity changes position. In fact, every observable that is a derivative of another observable is connected by quantum uncertainty to that other observable. It is impossible to remove that property of the universe from ‘programming’. At least in a sense, that the universe could work without it.”
The uncertainty in the position and velocity/momentum of the particle can be calculated by exactly first using position r and then momentum p, and then subtracting the product of momentum p and position r. Which can be used to obtain a difference in order, if any exists.
So the uncertainty in measurement is related to Planck constant h. If I replace position and momentum with any other observable which are derivatives of one another observable the same result is true and there is an uncertainty principle for the same observables. The expression I have written can be used to derive the Heisenberg uncertainty principle.
IF, for example:
then there would be no uncertainty in measurement and measuring first position and momentum or momentum and then the position would be the same physical process. But it is not the same for elementary particles, which where uncertainty comes from.
Electron as a wave Gaussian
In the vacuum, in free space the electron wavefunction can be mathematically obtained and the solution is called a Gaussian (after mathematician Gauss). It is a mathematical approximation of the point particle in space:
Instead of a point particle function with infinitely small length, Gaussian has a standard length of
which is used to describe the width of the Gaussian and probability of the electron being in a certain region.
The Gaussian can be represented as constituted of many waves that are centred on the initial position of a particle. The average velocity or momentum of the Gaussian at moment of measurement is zero. Free massive particle can have any energy and the energy is not quantized (if quantum gravity effects are not included).
Infinitely precise point particle function is a simplification, and it is not possible to localize particle to an infinitely dense region as the charge would be infinite, and then also energy and mass.
The average position of the electron is then
the electron and the Gaussian is centred on the initial position.
Probability of the electron being at a certain position in space is
The Gaussian is a special function, and the only function for which Heisenberg uncertainty principle is exactly true
The Gaussian wavefunction provides the most precise measurement with the smallest uncertainty possible allowed by quantum theory and nature. For example there is 68% probability that the electron will be in the region with width of one standard length. When measured, the wavefunction collapses and probability of the electron of being in certain region of space is described by the Gaussian.
“One of the reasons that it is impossible to completely understand or measure a quantum system in its entirety without uncertainty, is because we are made of the same quantum particles we are trying to measure. Another, is that universe is fundamentally probabilistic.”
Another unique special property of the Gaussian: if the electron is a Gaussian as a function of position, it is also a Gaussian as a function of momentum. Other wavefunctions do not have the same property. The momentum wavefunction describes an elementary particle without motion, but with small uncertainty in velocity.
If the width of the wavefunction (or uncertainty) in position space is small (so that position is precisely defined), the width of the particle function in velocity space will be large — so the uncertainty for the momentum/velocity will be large. Also, the opposite is true, if uncertainty (width of the Gaussian) in position space is large, momentum will be precisely defined.
Measurement by a photon, localizes the electron to a width defined by photon wavelength. So the more energy the photon has, the smaller the wavelength, and the electron is more precisely defined. So as a result, the uncertainty for velocity is large, and the electron will gain a random change in velocity for the measurement.
The Heisenberg microscope is a perfect microscope that can possibly be built in theory or in practice, by the most advanced technology. It is conceived to be technologically perfect, because it uses only one photon to observe only one particle. How does the measurement proceed?
After measurement of the Gaussian
Collapse of the wavefunction is usually described mathematically with a Gaussian function — in many instances experimentally the electron has a Gaussian probability in space. In a harmonic potential the particle has a Gaussian wavefunction.
What kind of properties will the electron have after the measurement by a photon depends on the photon used to observe the particle. As an effect after measurement the electron will receive a random change in velocity because of the measurement and collapse of the wavefunction.
The electron in the experiment is initially stationary, with average velocity zero. I present simulation of the Heisenberg microscope measurement.
Demonstration of motion of particle after measurement
In the first case the photon that observes the electron has small energy and large wavelength, then the electron after being measured gains a small velocity boost in random direction, because of uncertainty. Photon is observed and is determined experimentally that the electron position is determined by the large wavelength. The energy imparted to the electron is small.
Now the photon that observes the electron has large energy and small wavelength, then the electron after being measured receives a large velocity boost in random direction, because of uncertainty. Photon is observed and is determined experimentally that the electron position is determined by the small wavelength. The uncertainty in energy imparted to the electron is large.
The spread in the wavefunction had been neglected in the simulation.
Demonstration of uncertainty in space of a particle after measurement
After the measurement the electron will also change the width as the spread of the wavefunction in space.
For the next example, electron was measured with a precise photon and Gaussian was contained to a smaller region, after which there was large uncertainty in velocity — which resulted with position wavefunction spreading in space.
If the photon has large energy and small wavelength, and the electron is being measured along the y-axis, uncertainty in the momentum of the electron along the y-axis will be large, and the wavefunction in space will have a large spread. Along the x-axis nothing changes, because the particle properties were not measured along the x-axis. The particle is in a harmonic potential so there are oscillations around the center in the y-axis.
Conclusion
The ultimate conclusion is that the immediate future properties of the measured elementary particle depend on what kind of photon is used to perform the measurement (or some other particle that contains information about the observed particle).
The measurement itself is part of the reality. If the particle is observed with a different photon, the experiment will be different and then the measurement as well — so reality will change depending on what kind of measurement we perform on physical states. The Heisenberg uncertainty is not caused by inadequacy of our technology, but is inherent to the laws of our universe.
