Quantum Mechanics

The profession of auto mechanics originated in the late 19th and early 20th centuries alongside the development of the automobile industry. Mechanics initially focused on maintaining and repairing early car models. Over time, as vehicles became more complex, the field evolved to encompass various specialties and disciplines within automotive technology.

Harmonic perturbation refers to a periodic external force or disturbance applied to a system that is close to its natural harmonic frequency. This perturbation can affect the behavior of the system, causing resonance or other dynamic responses that are not present in the absence of the perturbation. Understanding and analyzing harmonic perturbation is important in various fields such as physics, engineering, and biology.

Dimensions in general represent degrees of freedom in motion. That means each dimension adds one independent direction in which an object may move.

Additional dimensions beyond the 3 basic and 1 time dimension work in the same way. However I suspect the question originates from the added dimensions that string theory (and other theories) proposes.

In that case it is important to note that these additional dimensions are curled up in a complex structure known as a Calabi-Yau manifold. These structures are very small and therefore cannot be seen with the eye.

A good example given by Brian Greene in one of his books is that of a garden hose which has ants walking on it. The hose represents a two dimensional space, one in the direction of the hose, and one which moves the ant around it. Close up you can clearly see that you have these two degrees of freedom.

However if you move far away the hose will appear to be a line and the second (curled up) dimension appears to vanish. In the same way the curled up dimensions that string theory predicts are also invisible to the naked eye, although they might eventually be probed by particle collider experiments.

Electrons are not formed from energy quanta; they are elementary particles that exist as fundamental units of matter. However, electrons can be created in processes such as beta decay, where a neutron transforms into a proton, electron, and an antineutrino. These processes involve the conversion of energy into matter, following the principles of quantum mechanics.

Classical mechanics is the alternative to quantum mechanics. It is a branch of physics that describes the motion of macroscopic objects using principles established by Isaac Newton. Unlike quantum mechanics, classical mechanics assumes that objects have definite positions and velocities at all times.

There is no reasonable alternative to quantum mechanics, at least not something that can even compare with the predictive power and experimental accuracy as quantum theory. If you want to make predictions about things happening at small scales you cannot do without quantum mechanics.

Also note that certain models which are now considered as possible theories of everything (e.g. string theory) all expand upon quantum mechanics, they do not make quantum mechanics invalid or unnecessary.

The merger of quantum mechanics with the special theory of relativity is commonly known as quantum field theory. According to this theory every particle of matter is just an excitation of a field that is everywhere in space. There is a different field for every different particle (this is not really true, but close enough).

You might already be familiar with electromagnetism. In that theory (which has now been completely absorbed into quantum field theory) electric and magnetic forces are transmitted via photons. These photons are just excitations of the photon field.

A photon has no mass, but all particles can be thought as as being excitations of fields. There is for example an electron field, but also a neutrino field and a muon field.

In the dirac view of quantum mechanics, operators are the center of analysis. An operator is some mathematical operation that acts on the wavefunction (psi) which returns an observable. Lets look at some examples:

say psi=exp(ik(dot)r)*exp(iomega*t) (which is the case for a free particle)

the momentum operator is the -ihbar gradiant applying this to our psi- we get hbar k. This is called the observable.

perhaps more familiar the energy operator which would likewise return hbar omega. Now doesn't that look familiar!

Interestingly enough, these two examples point out that the conservation of momentum and energy stem from the laws of physics being invariant, regardless of position and time.

Math Prelude:

Orthogonal wave functions arise as a natural consequence of the mathematical structure of quantum mechanics and the relevant mathematical structure is called a Hilbert Space. Within this infinite dimensional (Hilbert) vector space is a definition of orthogonal that is exactly the same as "perpendicular" and that is the natural generalization of "perpendicular" vectors in ordinary three dimensional space. Within that context, wave functions are orthogonal or perpendicular when the "dot product" is zero.

Quantum Answer:

With that prelude, we can then say that mathematically, the collection of all quantum states of a quantum system defines a Hilbert Space. Two quantum functions in the space are said to be orthogonal when they are perpendicular and perpendicular means the "dot product" is zero.

Physics Answer:

The question asked has been answered, but what has not been answered (because it was not was not asked), is why orthogonal wave functions are important. As it turns out, anything that you can observe or measure about the state of a quantum system will be mathematically represented with Hermitian operators. A "pure" state, i.e. one where the same measurement always results in the same answers, is necessarily an eigenstate of a Hermtian operator and any two pure states that give two different results of measurement are necessarily "orthogonal wave functions."

Conclusion:

Thus, there are infinitely many orthogonal wave functions in the set of all wave functions of a quantum system and that orthogonal property has no physical meaning. When one identifies the subset of quantum states that associated pure quantum states (meaning specifically measured properties) and then two distinguishable measurement outcomes are associated with two different quantum states and those two are orthogonal. But, what was asked was a question of mathematics. Mathematically orthogonal wave functions do not guarantee distinct pure quantum state, but distinct pure quantum states does guarantee mathematically orthogonal wave functions. You can remember that in case someone asks.

The smallest particle in quantum physics is typically considered to be the quanta of energy known as a photon, which is a fundamental particle that carries electromagnetic radiation. However, there are also other elementary particles such as electrons, quarks, and neutrinos that are considered fundamental building blocks of matter.

The speed of light was predicted before it was ever measured. If you write the

differential equation of a wave, there's a very definite place in the equation where

the speed of the wave appears.

The Scottish Physicist James Clerk-Maxwell took four equations of electricity and

magnetism that had been discovered by earlier scientists, and succeeded in

mashing them together to come up with the equation of an electromagnetic wave.

Right there in the spot where the wave speed should be was the quantity

[ (electrostatic permittivity of space) x (magnetic permeability of space) ] .

Both of those properties of space had been kind-of measured by Maxwell's time,

but now it suddenly became very urgent to go back and work on measuring them

with the utmost accuracy ... and at the same time, to figure out a way to measure

the speed of light. Because if that quantity could be shown to match the real speed

of light, then we would know that light is an electromagnetic wave, and that Maxwell's equation for it is an accurate mathematical description of it.

You asked "Why is that the speed of light ?" The answer is: Because light is an

electromagnetic wave, therefore its speed is determined by those two properties

of free space, and the values of those two properties of space happen to be

such and such."

Now, we can almost hear you asking "Why are the values of those two characteristics

of space the numbers that they are, and not a bit more or less ?"

All we can tell you is that there are cosmologists who are actually working on that

esoteric question, along with the question of why other properties of the universe (For example, the gravitational constant.) are the numbers that they are. One realization that has emerged is the fact that if a few of the constant numbers of nature were just slightly different from what they actually are, then life would not be possible.

As the wavelength increases to infinity the electro-magnetic continuum take on a new base value and with no variation has no radiation to transmit.

As the wavelength decreases to zero the energy packet become a massive body and therefore is no longer a radiating.

It has to do with probabilities. The area under the curve of a wavefunction can be whatever you want it to be. You normalize the curve to have the total probability equal to 1, which makes the mathematics a lot easier. We do this with statistics and probabilities all the time.

The correct spelling is 'precise mechanics.' 'Precise' means exact or accurate, so precise mechanics refers to detailed and accurate mechanics in a specific context.

We know how big it is now, we know how old it is, and we know the rate of expansion. Its like rewinding a movie that you started in the middle. We just rewind the known universe as it is now. We dont know how big it was before the big bang but we know how big it was at the initial kaboom.

I'm not sure what your question is asking, but I can try to give an answer.

The rotation of molecules, for example, are quantized at the quantum scale. We can use the rigid rotor model from classical physics to help describe the potential part of the Hamiltonian operator, as well as the form of the wave equation needed to find the energy of a particular rotational state.

It would be similar to using the simple harmonic oscillator to model the potentials and wavefunctions needed needed calculate the energy of vibrational levels of a molecule.

John von Neumann achieved his goal through his groundbreaking work in mathematics, physics, and computer science. He made significant contributions to game theory, quantum mechanics, computer architecture, and the development of the atomic bomb. Von Neumann's ability to think abstractly and apply his insights across multiple disciplines helped him achieve success in his pursuits.

The uncertainty principle states that there is a fundamental limit to how precisely we can know certain pairs of properties of a particle. While this concept might not directly impact our daily lives in obvious ways, it underpins our understanding of the behavior of particles at the quantum level, which has implications for technology, such as in the development of quantum computing and modern electronics.

Matter can only 'be'. It's existence is only an idea. For the convenience of analysis, we can think that matter 'exists'. To exist, it needs not an idea but the fourth dimension, that is time. Something that has length, breadth and height can not be said to have existed, if it had not gone through a duration of at least a split-second. So matter can exist without ideas, but can not without duration.

FBD stands for Free Body Diagram. In mechanics, a Free Body Diagram is a visual representation of an object with all the external forces acting on it shown as vectors. It helps in analyzing the forces acting on the object and determining its motion or equilibrium.

Although determining the cell constant accurately is ideal for precise measurements, it is not always necessary before conducting the experiment. As long as the conductance cell is properly calibrated, any variations in the cell constant can be accounted for in the calculations. The key is consistency in the calibration process to ensure reliable and reproducible results.

The equation, as originally written by Erwin Schrodinger, does not use relativity. More complicated versions of his original equation, which do incorporate relativity, have been developed.

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The "fermion sign problem" refers to a computational challenge in quantum many-body problems when simulating systems of fermions using Monte Carlo methods. As fermions follow the Pauli exclusion principle, the wavefunction must be antisymmetric under particle exchange. This can lead to an exponentially growing number of configurations needed to accurately simulate the system, making calculations computationally demanding or impractical.