Quantum state

The quantum state is the physical state that a physical system has at a given moment in the framework of quantum mechanics. In classical physics, theoretically, when measuring a physical quantity in a system several times, we would obtain the same value. However, in quantum physics, in theory, when measuring a physical magnitude we could obtain a different value each time it is measured. Therefore, to study the results of a quantum measurement, a probability distribution is used.

Introduction

Quantum physics is a physical theory in which the measurement process is not deterministic, this means that given two physical systems with the same quantum state, when measuring a certain magnitude on them, the same value does not have to be obtained. This is in stark contrast to the notion of measurement in classical mechanics. Quantum mechanics is a theory that accounts for the probabilistic nature of the measurement process and both its formalism and the notion of quantum state are abstractions in order to explain the experimental fact of measurement indeterminacy.

In the quantum mechanical formalism, physical systems are represented mathematically by a state vector for pure states or by a density matrix for mixed states. Equivalently, the state vector is also representable as a wave function (in continuous basis representations). Both the state vector and the density matrix allow predicting possible values of the experiments associated with the measurement of physical observables.

The quantum state is an abstract mathematical representation, so there is a source of difficulty in dealing with this formalism of the theory for the first time since there are no good intuitive classical analogues. Especially, that the quantum state is not the state in which it can be found, since when observing a quantum object an eigenvalue for that observable is always obtained, even though the system state is not an eigenstate for that observable.

Examples

Particle in a bound and spinless state

Given a small particle, whose presence is limited to a fairly localized region of space, such as an atom electron, its quantum state can be adequately represented by a wave function. In that case the quantum state is a square integrable function defined in all three-dimensional space. Naturally the function will only take values significantly different from zero in a region around the atomic nucleus of the approximate size of the atom. The module of said squared function is associated with the probability density of finding the particle at a certain point, in such a way that:

Pr(V)=∫ ∫ V日本語END END (x)日本語2d3x{displaystyle mathrm {Pr} (V)=int _{V} organopsy (mathbf {x})int^{2}mathrm {d} ^{3}mathbf {x}}}}

The set of all functions that can potentially represent the quantum state of an electron in an atom constitutes a vector space of infinite dimension. The interest of this space of functions is that it allows defining linear operators that represent the effect of a possible measure, thus the average value of a possible measure is given by:

MEND END =∫ ∫ R3END END ↓ ↓ (M^ ^ END END )d3x{displaystyle langle M_{psi }rangle =int _{mathbb {R}{3}{3}psi ^{*}({hat {M}}psi)mathrm {d} ^{3}mathbf {x} }}

while the possible values for the same magnitude coincide with the spectrum of the operator. The probability distribution of the different values is given by the third postulate of quantum mechanics.

A bound state is a quantum state of a physical system that is a linear combination of stationary states corresponding to values of the energy of the point spectrum of the system's Hamiltonian.

Particle in a collision state

The mathematically precise definition of the unbound state is complex. Intuitively, a particle that executes a movement in a finite region of spacetime or that with probability one is located in a finite region is a bound state. Collision states are unbound states and therefore lack those properties. The simplest example of a collision state is a particle with a perfectly defined momentum, whose state can be represented by a plane wave.

A state of collision or unlinked state, is a quantum state such that the extent of probability is not annulled outside any finite region of physical space (nor does it decay exponentially or evenly outside any finite region). The collision states therefore represent particles that can move through an infinite region of space and that whose wave function also does not fall abruptly to zero (in an exponential way). A particle without a spine with a perfectly defined moment p=(px,pand,pz){displaystyle mathbf {p} =(p_{x},p_{y},p_{z}}}}} has a representative state by function:

END END (x,and,z)=ei(pxx+pandand+pzz)/ {displaystyle psi (x,y,z)=e^{i(p_{x}x+p_{y}y+p_{z}z)/hbar }}}

Note that this function, like those representing many other collision states, is not a normalizable function (that is, square integrable) and therefore cannot be represented as an element of an ordinary Hilbert space. In order to be able to treat collision states rigorously within a formalism similar to that of ordinary Hilbert spaces, equipped Hilbert spaces were introduced, where the collision spaces are dual elements of a certain nuclear subspace of said Hilbert space..

Collision states are widely used in quantum field theory and particle physics to represent particle collision experiments. In many of these experiments, the interaction between two types of particles takes place in a relatively small and localized region of space; outside of that region where the interaction occurs, the particles move freely without interaction and therefore are unbound states that can perform an unbounded motion, and for that reason they are represented as non-renormalizable collision states (where the presence probability amplitude does not decay to zero).

Pure multi-particle state

The spin-statistics theorem implies that the quantum state of a system of indiscernible (and therefore identical) particles must be an eigenstate of any particle exchange operator. Since these operators are idempotent they only admit +1 or -1 as eigenvalues and therefore any physically realizable state must be symmetric or antisymmetric with respect to the exchange of any two particles. The spin-statistics theorem further proves that an indiscernible fermion state must be an antisymmetric state while an indiscernible boson state must be symmetric.

Mixed state of various particles

State Consistency

The more effect-free the situation (as in the case of Schrödinger's cat experiment), the more quantum the system.

In simpler words, the quantum state is one in which the atom is completely free from any interaction with variables that can change its pure state, be it light, heat, or any other interaction, and with the interaction strongly perturbs the system, that is, the quantum effects disappear. The process by which this disturbance produces the loss of some characteristics of typically quantum behavior is known as quantum decoherence.

Dirac notation

Dirac invented a powerful and intuitive notation to capture this abstraction in a mathematical tool known as the bra-ket notation. This is a very flexible notation, and allows very suitable formal notations for theory. For example, it can refer to a LICexcited atom▪ 日本語↑ ↑ {displaystyle Δ!uparrow rangle } for a system "with spin up", or even to 日本語0 {displaystyle Δ0rangle } and 日本語1 {displaystyle Δ1rangle } when dealing with qubits. This conceals the complexity of the mathematical description, which is revealed when the state is projected on a coordinate basis. For example, the compact notation Δ1s HCFC, which describes the hydrogenoid atom, becomes a complicated function in terms of Laguerre polynomials and spherical harmonics by projecting it at the base of the Δ position vectors.r. The resulting expression (r)=rأعر, known as wave function, is the spatial representation of the quantum state, specifically its projection in the real space. Other representations are also possible, such as the projection in the space of moments (or reciprocal space). The different representations are different facets of a single object, the quantum status.

Superposition of pure states

The superposition of pure states is that superpositions of them can be formed. Yeah. 日本語α α {displaystyle Δalpha rangle } and 日本語β β {displaystyle Δbeta rangle } are two kets that correspond to quantum states ket

cα α 日本語α α +cβ β 日本語β β {displaystyle c_{alpha }{alpha rangle +c_{beta Δ}{beta rangle }

is a different quantum state (possibly not normalized). Bearing in mind that the quantum state depends on the amplitudes and phases (arguments) of cα α {displaystyle c_{alpha }} and cβ β {displaystyle c_{beta }. In other words, for example, even though 日本語END END {displaystyle 日本語psi rangle } and eiθ θ 日本語END END {displaystyle e^{itheta } (θ θ {displaystyle theta } being real) corresponds to the same physical quantum state, they are not interchangeable, as, for example, 日本語φ φ +日本語END END {displaystyle Δphi rangle +ALESpsi rangle } and 日本語φ φ +eiθ θ 日本語END END {displaystyle Δphi rangle +e^{itheta } in general, it corresponds to the same physical state. However, 日本語φ φ +日本語END END {displaystyle Δphi rangle +ALESpsi rangle } and eiθ θ (日本語φ φ +日本語END END ){displaystyle e^{itheta }(ATAphi rangle +ALESpsi rangle)} if they correspond to the same physical condition. This is sometimes described by saying that "global" phase factors are not physical.

An example of a quantum interference phenomenon arising from superposition is the double-slit experiment. The photon state is a superposition of two different states, one of which corresponds to the photons having passed through the left slit and the other corresponding to the right slit. The relative phase of the two states has a value that depends on the distance of each of the two slits. Depending on what the phase is, the interference is constructive in some places and destructive in others, creating the interference pattern.

Another example of the importance of relative phase in quantum superposition is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. The resulting overlay ends up oscillating between two different states.

Degenerate and non-degenerate states

For many physical systems, for each value of energy there is only one possible state of the system, in which case the states of said system are called non-degenerate. However, in other systems for some energies there is more than one possible state with that energy. When there is more than one possible quantum state for a given energy, each of the possible states is called a degenerate state. The level of degeneracy is the number of possible states.

An example of a quantum system that presents degenerate states is the hydrogenoid atom in which each energy level of the atom can house two electrons of the same energy, that is, each electron can be in one of the two possible states for that level, and therefore both states are degenerate states. In Schrödinger's atomic model the degeneracy is 2n² since all quantum states that share the principal quantum number n and the quantum number azimuthal l have the same energy, and there are 2n² possible states for the same energy. If the relativistic corrections are taken into account, the Dirac atomic model is obtained where, due to said corrections, the states with different azimuthal quantum number l have different energies, and therefore there are only 2(2l+1) (< 2n²) states with the same energy (all those that share in magnetic quantum number. If additionally subjected to the atom to a magnetic field, the degeneracy is completely eliminated by a doubling of the energy levels, now each electron having slightly different energy and now existing a one-to-one relationship between possible energies and possible states.

Examples

It is instructive to consider the most useful quantum states of the quantum harmonic oscillator:

• The state of Fock Δn▪n whole number) that describes a defined state of energy.
• The coherent state Δαγ (α number complex) that describes a definite phase state.
• The thermal state that describes a state in thermal balance.

The first two states are pure quantum states, that is, they can be described by a "ket" Dirac state, while the latter is a quantum mixed state, that is, a statistical mixture of pure states. A mixed state needs a statistical description in addition to the quantum description. This is achieved with the density matrix, which extends quantum mechanics to statistical quantum mechanics.