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eigenstate
(quantum mechanics) A dynamical state whose state vector (or wave function) is an eigenvector (or eigenfunction) of an operator corresponding to a specified physical quantity. energy state.
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What is the spin-helix state?
The momentum independent eigenstate defined for a twodimensional electron gas withlinear in momentum Bychkov-Rashba and Dresselhaus type spin-orbit interaction of equal magnitude. In momentum space this state is characterized by a +pi/4 or -pi/4spin orientation in the plane of the electron gas.
Why do carbon share electrons?
Because the eigenfunction of the collapsed wave function that results from inserting the Hamiltonian of C2's composite orbital's degrees of freedom into Schrodinger's equation yielded a lower eigenstate for that situation than if you did the similar thing to an unbounded carbon atom. Woof! I couldn't have said it better myself. How refreshing! I can just hear Mortimer Snerd squeezing out a yowl. That was beautiful!
Why do 2 electrons which have the same charge not repel each other?
Electrons repel because they are charged and there is Coulomb repulsion between them. This is not a phenomena that depends on quantum mechanics or the fact that they are moving in the same direction. Electrons repel when they are moving in any direction or not moving at all. The quantum mechanical description of the time evolution of a two particle system will depend on the interaction between the two particles. It also depends on the spin and statistics of the particles. In this case they are two identical fermions with a Coulomb repulsion and unspecified spin. It has also not been specified whether the quantum state in question is an eigenstate of the Hamiltonian. There is not much more to say other than one needs to be clear as to whether one is asking a question about the Hamiltonian or the wave function.
Why do some isotopes decay while others do not?
Some isotopes are unbalanced in certain ways, and some have unnecessary, extra energy. Decaying lets these isotopes balance themselves out to a more-stable, lower-energy state. Once a certain state of energy is hit, it becomes energetically unfavorable for the atom to decay further, so they don't. All three main types of radioactive decay can be thought of as stabilizers. Alpha decay stabilizes nuclear size: As nuclei get bigger and bigger, electromagnetic repulsion between the protons begins to overtake the strong nuclear force's hold on the nucleons, so the isotope spits out a helium nucleus to compensate. Beta decay stabilizes the ratio of protons to neutrons: This is done by effectively turning a neutron into a proton through the emission of an electron. This is needed because neutrons by themselves are unstable so they have to be constantly exchanging a particle called a gluon with the surrounding protons to stick together. If there's too many neutrons, some won't be able to do this. Gamma radiation stabilizes the overall nuclear energy: They do this by reducing the nucleus to a lower energy eigenstate through the release of a high-energy photon.
What are orthogonal wave functions?
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.
What are some songs in physics?
Bohemian Mechanics by Jake Ralston (goes to the tune of Bohemian Rhapsody by Queen)Is this the real life- Is this just classical- Caught in a square well- No escape from potential walls- Open your mind To quantum mechanics and see- The particle's in, a superposition- Of many eigenstates, wave functions, Some spin up, some spin down, Phipps and Taylor's results, tells us spin's intrinsic, to e- To e- Teacher, just cut a tip, Put it in the S.T.M. Saw the tunneling current, Teacher, the tip worked so well, But now I've gone and dropped it on the floor- Teacher, ooo-, Didn't mean to call Suzanne- But she's the only one who cuts it well- Overflow, overflow, cause the current's way too high. Wavestate, time to collapse, Become an eigenstate- Of the measurement you take, Goodbye, operator- I've got to change- Gotta get inside the bra- and the ket Commutator ooo- (depends on which one goes first) You will never know, Two things when the commutator's not zero. I see a result which implies a paradox, Think it through, think it though will you do the thought experiment- Fermi calculations- very, very frightening me- Richard Feynman, Richard Feynman, Albert Einstein, Albert Einstein, Richard Feynman Q.E.D. - Relativity-- I'm just a proton with an electron- He's hydrogenic, thus fully solvable! Spare him his pride from approx-imations! Spherical, Cartesian- will you let me choose? Cartesian! No-, we must choose spherical- Let me choose- Cartesian! We must choose spherical- Let me choose- Cartesian! We must choose spherical- Let me choose- Must choose spherical- Let me choose- Must choose spherical- Let me choose- Rho, rho, rho, rho, rho, rho, rho- Sphere harmonics, sphere harmonics, sphere harmonics, superpose! Sch-rödinger has a value put aside for me, for me, for me! So you think you can measure time and energy? So you think you can measure Ly and Lz? Oh baby- Heisenberg says no baby- There's uncertainty, there's uncertainty built in here. Nothing's really matter, E is m c squared, Nothing's really matter, nothing's really matter to me. (Anyway the spin goes...)
How would you know the hypothesis for growing plants?
this: where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x. The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution. The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. During a measurement, on the other hand, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e., random. A time-evolution simulation can be seen here. Wave functions can change as time progresses. An equation known as the Schrödinger equation describes how wave functions change in time, a role similar to Newton's second law in classical mechanics. The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates. Some wave functions produce probability distributions that are constant, or independent of time, such as when in a stationary state of constant energy, time drops out of the absolute square of the wave function. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus. The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the wave-like behavior of quantum states. It turns out that analytic solutions of Schrödinger's equation are only available for a small number of model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion and the hydrogen atom are the most important representatives. Even the helium atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of quantum chaos. There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the transformation theory proposed by Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom). An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the quantum-mechanical counterpart of action principles in classical mechanics. cheers!
