Quantum Mechanics

Mathematically, it's based upon analysis (calculus, differential equations, etc.) as are most fields of physics, but what really sets quantum apart from other physics fields is it's fundamental mathematical dependence on probability, linear algebra, and group theory.

It's conceptually based on uncertainty, probability, and discreteness (as opposed to absolute and continuous).

It doesn't, from the equation E = h*f (E is energy, h is Planck's constant, f is frequency) you can clearly see that energy is a function of frequency, not amplitude (intensity). Therefore, it doesn't even matter what the relationship between stopping potential and energy is, because it will only depend on frequency, which is sufficient knowledge to answer this question.

Light is narrow on the spectrum because it consists of electromagnetic waves with specific wavelengths. Each color of light corresponds to a different wavelength, resulting in the distinct colors we see in the visible spectrum. This narrow range is due to the specific interactions between photons and atoms/molecules that determine the wavelengths of light that are emitted or absorbed.

Einstein's work on the Photoelectric effect, which won him the Nobel prize in 1921 was a bulwark of Quantum Mechanics. Einstein went off in another direction because of his inate suspicion that Quantum Mechanics has severe internal difficulties. Quantum Mechanics and Relativity have not yet been reconciled--but they are moving together slowly. Quantum Gravity seems to be key to the issue and may be resolved by String Theory.

For general waves...probably d'Alembert, who solved the one-dimensional wave equation. In quantum it would have to be Schrodinger.

Light has have wavelengths across the electromagnetic spectrum. Very little of this spectrum can actually be seen. So it behaves like a wave in that it can be decoded in the case of visible light by your eyes and turned into a useful information by your brain. Radio waves, which are lower on the spectrum than visible light, is decoded by radios. So we know it is a wave. What we also know is that it is a particle. Electrons can be harvested into electricity by PV cells so we also know that it is particle too. There is a fair amount of study into light and the properties of light.

The uncertainty principle was formulated by German physicist Werner Heisenberg in 1927 as part of his work in quantum mechanics. It states that certain pairs of physical properties, such as position and momentum of a particle, cannot be precisely known simultaneously.

What's meant is that the equation has multiple solutions all existing at the same time. In quantum, the observable solution of a state isn't determined until measurement, before that, the state is simultaneously in all solutions.

Phase difference in wave propagation results in interference patterns. When waves with a phase difference interact, they can either reinforce each other (constructive interference) or cancel each other out (destructive interference), affecting the overall amplitude of the resulting wave. This phenomenon is commonly observed in various wave systems, such as light and sound.

The function of damping current is to reduce oscillations or ringing in a circuit by dissipating excess energy. It helps stabilize the system and prevent it from overshooting or oscillating uncontrollably. Damping currents are often used in applications like electrical circuits, mechanical systems, and control systems to improve system response and stability.

No, Schrödinger's cat is a thought experiment to illustrate the concept of quantum superposition. It has not been practically attempted due to ethical concerns and technological limitations in creating such a scenario where a cat could be in a superposition of alive and dead states.

J mesons are subatomic particles that do not experience a melting phase transition like larger particles or materials. As such, they do not require energy to melt as they do not solidify.

Because every material has unique allowed energy levels, the electromagnetic radiation emitted from them, which is equal to the energy of the higher excited state minus the energy of the lower excited state (or ground state) are also going to be unique. Since color is based on an electromagnetic wave's frequency, which in turn is proportional to energy, these unique waves give unique colors.

In thermodynamic equilibrium, the system's entropy is maximized, reaching a state of maximum disorder or randomness. This is unique compared to other states of the system where entropy may be increasing or decreasing as the system approaches equilibrium. At equilibrium, the system has reached a stable condition where the distribution of energy and molecules is uniform, making it a distinct state in terms of entropy.

Quantum entanglement cannot be used to transport energy from one place to another. While entangled particles exhibit a strong correlation that allows for instantaneous changes in one particle to be reflected in the other, this correlation cannot convey energy or information faster than the speed of light. Transporting energy still requires physical processes and mechanisms.

The quantum theory of light unifies the particle theory of light (photons) and wave theory of light by treating light as both particles and waves. Photons are quantized packets of energy that exhibit particle-like behavior, while light waves exhibit wave-like behavior with properties such as interference and diffraction. Quantum theory provides a framework to understand the dual nature of light.

In probability theory, an "expectation value" is the average of all values of a measurable quantity that one would expect, if a measurement was repeated a large number of times on a given system. For example, for an unbiased coin, the expectation value for "heads" is half of all tosses.

Each measurable quantity of a quantum system has an operator that, when mathematically applied to the system, gives a value of that quantity for that system. The expectation value for that quantity, for a given quantum system, is the product of that operator on a given state of the system, times the probability of the system being in that state, integrated over all possible states of the system. A more formally stated example:

For a quantum state Ψ(x), where 'x' can vary from -∞ to ∞, and for which Q(x) is a measurable quantity, then the expectation value of Q(x) would be equal to

∫Ψ*(x)Ψ(x)Q(x)dx

integrated from x = -∞ to x = ∞

As an example, suppose we wanted the expectation value for the radial position of an electron in its '1S' state within a hydrogen atom. When doing the formal math, we find that this value exactly equals the Bohr Radius. In contrast to the Bohr Model of an atom, this expectration value does NOT state that this electron IS at this radius, only that an AVERAGE of all radial measurements of such an electron would be the Bohr Radius.

Entangled subatomic particles do not involve conveying information faster than light because the act of measuring one particle's state instantaneously determines the state of the other particle, regardless of the distance between them. This correlation is a result of the shared quantum state between the particles at the time of entanglement, not a form of communication. The information remains random and cannot be controlled to send a message.

A wave function is normalized by determining normalization constants such that both the value and first derivatives of each segment of the wave function match at their intersections.

If instead you meant renormalization, that is a different problem having to do with elimination of infinities in certain wave functions.

Nobody is really quite sure yet. The existence of the Higgs boson is predicted by the Standard Model of quantum mechanics, but nobody has yet been able to experimentally detect one, so a lot of the details of it are still unknown.

The Standard Model does not predict what mass the Higgs boson would have, so it could be anything, really, though it's generally assumed that its mass is somewhere between 115 and 180 GeV/c2, because if it is that will make all the equations we have work properly for pretty much all cases. It is possible, however, that we'll find out that it isn't in this range (or we may not ever be able to find one at all), in which case people may have to make some changes to our current theories to account for why it's different than we expected.

Exchange degeneracy in quantum mechanics refers to the phenomenon where multiple particles with the same properties (such as electrons in an atom) are indistinguishable from each other, leading to the degeneracy of energy levels. This occurs due to the symmetric nature of the wavefunctions describing the particles, which do not change if the particles are exchanged. Exchange degeneracy plays a crucial role in determining the structure and properties of atoms, molecules, and other quantum systems.

Hydrogen is special in quantum mechanics because its simplest form, the hydrogen atom, is the only atom for which the Schrödinger equation can be solved analytically. This allows for detailed insight into the behavior of electrons in the atom, providing a fundamental understanding of quantum mechanics. Additionally, hydrogen plays a key role in the development of quantum theories and helps explain important phenomena such as emission spectra and energy levels in atoms.

By definition a massless particle has no rest mass therefore it can not take up any spacial volume. I think the confusion lies with calling something that is massless, a particle. This is because as soon as we hear particle we think "object" and objects have definite mass and volume. A photon is massless and sometimes people may refer to it as a particle of light. But in fact that is sort of a misnomer being that it really isn't a particle, though it has particle-like properties. If something is massless theorists have said that the object does not interact with the Higgs field, though gravitational effects are still felt by the photon, example: gravitational lensing.