Schrodinger equation

The Schrödinger equation , developed by the Austrian physicist Erwin Schrödinger in 1925, describes the time evolution of a quantum subatomic particle with mass in a non-relativistic context. It is of central importance in the theory of ordinary quantum mechanics, where it plays a role for microscopic particles analogous to Newton's second law in classical mechanics. Microscopic particles include elementary particles, such as electrons, as well as systems of particles, such as atomic nuclei.

It is very important in many applications, although in relativistic contexts it should be replaced by the treatment of quantum field theory, since the Schrödinger equation does not contemplate the processes of pair creation or particle annihilation.

Equation

Time Dependent Equation

The form of the Schrödinger equation depends on the physical situation. The most general form is the time-dependent equation, which describes a system that evolves over time:

A wave function that satisfies Schrödinger's non-relativist equation with V = 0. That is, it corresponds to a particle freely traveling through the free space. This chart is the real part of the wave function.

Time-dependent Schrödinger equation (general) i ▪ ▪ ▪ ▪ t (r,t)=H^ ^ (r,t){displaystyle ihbar {frac {partial }{partial t}}{Psi (mathbf {r}t)={hat {H}Psi (mathbf {r}t)}}}

where i is the imaginary unit, ħ is the "reduced Planck's constant" or "Dirac's constant" (Planck's constant divided by 2π), the symbol ∂/∂t indicates a partial derivative with respect to time t, Ψ (the Greek letter psi) is the wave function of the quantum system, and Ĥ is the operator Hamiltonian differential (which characterizes the total energy of any given wave function and has different forms depending on the situation).

Each of the three rows is a wave function that satisfies the time-dependent Schrödinger equation for a quantum harmonic oscillator. To the left: The real part (blue) and the imaginary part (red) of the wave function. Right: The distribution of probability of finding a particle with this wave function in a given position. The two rows above are examples of stationary statesThat corresponds to stationary waves. The bottom row is an example of a state that No. It's stationary. The column on the right illustrates why the state can be called "stationary."

The most famous example is the non-relativistic Schrödinger equation for a simple particle moving in an electric field (but not in a magnetic field; see Pauli's equation):

Time-dependent Schrödinger equation (non-relativist simple particle) i ▪ ▪ ▪ ▪ t (r,t)=[chuckles]− − 22μ μ ► ► 2+V(r,t)] (r,t){displaystyle ihbar {frac {partial }{partial t}}}Psi (mathbf {r}t)=left[{frac {hbar ^{2}{2}{2mu }}nabla ^{2}+V(mathbf {r}tright]f

where μ is the "reduced mass" of the particle, V is its potential energy, ∇² is the Laplacian (a differential operator), and Ψ is the wave function (more precisely, in this context, it is called " position-space wave function"). That is, it means that the "total energy is equal to the kinetic energy plus the potential energy".

According to the differential operators that are used, it is observed that it is a differential equation in linear partial derivatives. It is also a case of a diffusion equation, but not like the heat equation, since it too is a wave equation given per imaginary unit present in the transient term.

The term "Schrödinger equation" can refer to the general equation (the first above), or the specific non-relativistic version (the second and its variants). The general equation is used throughout quantum mechanics, from Dirac's equation to quantum field theory, through the use of complicated expressions for the Hamiltonian. The specific non-relativistic version is a simplified approximation to reality, which is quite accurate in many situations, but very imprecise in many others (see relativistic quantum mechanics and relativistic quantum field theory).

To apply the Schrödinger equation, the Hamiltonian operator is used for the system, taking into account the kinetic and potential energies of the particles that make up the system, and then inserted into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system.

Equation independent of time

The time-independent Schrödinger equation predicts that wave functions can be in the form of standing waves, called stationary states (also called "orbitals", as in atomic orbitals or molecular orbitals). These states are important, and if the stationary states are classified and can be understood, then it is easier to solve the time-dependent Schrödinger equation for any state. The time-independent Schrödinger equation is the equation that describes steady states. (Only used when the Hamiltonian is not time dependent. However, in each of these cases the total wave function will still be time dependent.)

Schrödinger equation independent of time (General) E =H^ ^ {displaystyle EPsi ={hat {H}}Psi }

That is, the equation says that:

When the Hamiltonian operator acts on a certain wave function , and the result is proportional to the same wave function , then is a stationary state, and the constant proportionality, EIt's state energy. .

The time-independent Schrödinger equation, in linear algebra terminology, is an eigenvalue equation.

A well-known application is the non-relativistic Schrödinger equation for a simple particle moving in an electric (but not magnetic) field:

Schrödinger equation independent of time (simple non-relativist particle) E (r)=[chuckles]− − 22μ μ ► ► 2+V(r)] (r){displaystyle EPsi (mathbf {r})=left[{frac {-hbar ^{2}}{2mu }}}nabla ^{2}+V(mathbf {r})right]Psi (mathbf {r}}}}}}}

Source of the equation

Historical context

At the beginning of the centuryXX. It had been proved that the light had a duality of a corporeal wave, that is, the light could manifest (according to the circumstances) as a particle (photo in the photoelectric effect), or as an electromagnetic wave in the luminous interference. In 1923 Louis-Victor de Broglie proposed to generalize this duality to all known particles. He proposed the paradoxical hypothesis at the time that all classical microscopic particles can be assigned a wave, which was experimentally checked in 1927 when electron diffraction was observed. By analogy with photons, De Broglie associates with each free particle with energy E{displaystyle E} and amount of movement p{displaystyle p} a frequency .. {displaystyle nu } and a wavelength λ λ {displaystyle lambda }:

{E=h.. p=h/λ λ {displaystyle left{begin{matrix}E=hnu \p=h/lambda end{matrix}}right. !

The experimental verification made by Clinton Davisson and Lester Germer showed that the wavelength associated with the electrons measured in the diffraction according to the Bragg formula corresponded to the wavelength predicted by the De Broglie formula.

That prediction led Schrödinger to try to write an equation for the associated de Broglie wave that for macroscopic scales would reduce to the equation of classical particle mechanics. The classical total mechanical energy is:

E=p22m+V(r){displaystyle E={p^{2} over 2m}+V(r)}

The success of the equation, deduced from this expression using the correspondence principle, was immediate by the evaluation of the quantified energy levels of the electron in the hydrogen atom, since this allowed explaining the emission spectrum of hydrogen: series from Lyman, Balmer, Bracket, Paschen, Pfund, etc.

The correct physical interpretation of the Schrödinger wave function was given in 1926 by Max Born. Because of the probabilistic nature that was introduced, Schrödinger's wave mechanics initially aroused the distrust of some renowned physicists such as Albert Einstein, for whom "God does not play dice" and Schrödinger himself.

The historical derivation

The conceptual scheme used by Schrödinger to derive his equation rests on a formal analogy between optics and mechanics:

• In the wave optics, the equation of propagation in a transparent real index medium n slowly varying to the wavelength scale leads — while a monochromatic solution is sought where the amplitude varies very slowly in the face of the phase — to an approximate equation called eikonal. It is the approximation of the geometric optics, to which the Variational Principle of Fermat is associated.
• In the hamiltonian formulation of classical mechanics, there is an equation of Hamilton-Jacobi (which is ultimately equivalent to Newton's laws). For a non-relativist mass particle subjected to a force that derives from a potential energy, total mechanical energy is constant and Hamilton-Jacobi’s equation for Hamilton’s characteristic “function” formally looks like the equation of the eikonal (the associated variable principle is the minimum action principle).

This parallelism had already been noted by Hamilton in 1834, but he had no reason to doubt the validity of classical mechanics. After de Broglie's 1923 hypothesis, Schrödinger says: the eikonal equation being an approximation to the wave equation of wave optics, we seek the wave equation of "wave mechanics" (to be done) where the approximation will be the Hamilton-Jacobi equation. What is missing, first for a standing wave (E = cte), then for a wave of any kind.

Schrödinger had indeed begun by treating the case of a relativistic particle —like de Broglie before him—. He had then obtained the equation known today as the Klein-Gordon, but its application to the case of electric potential of the hydrogen atom gave energy levels incompatible with the experimental results. This will make him focus on the non-relativistic case, with known success.

Statistical interpretation of the wave function

At the beginning of the 1930s Max Born who had worked together with Werner Heisenberg and Pascual Jordan in a version of quantum mechanics based on alternative matrix formalism to Heisenberg's one appreciated that the complex Schrödinger equation has a integral movement given by END END ↓ ↓ (x)END END (x)(=日本語END END (x)日本語2){displaystyle scriptstyle psi ^{*}(x)psi (x)(=associatedpsi (x)associated^{2})}} that could be interpreted as a probability density. Born gave the wave function a probabilistic interpretation different from that given by De Broglie and Schrödinger, and for that work he received the Nobel Prize in 1954. Born had already appreciated in his work through the matrix formalism of quantum mechanics that the set of quantum states naturally carried to build Hilbert spaces to represent the physical states of a quantum system.

In this way, the approach of the wave function as a material wave was abandoned, and it began to be interpreted in a more abstract way as a probability amplitude. In modern quantum mechanics, the set of all possible states in a system is described by a separable complex Hilbert space, and any instantaneous state of a system is described by a "unit vector" in that space (or rather an equivalence class of unit vectors). This "unit vector" encodes the probabilities of the outcomes of all possible measurements made to the system. Since the state of the system generally changes with time, the state vector is a function of time. However, it must be remembered that the values of a state vector are different for different locations, in other words, it is also a function of x (or, three-dimensionally, of r). The Schrödinger equation gives a quantitative description of the rate of change in the state vector.

Modern formulation of the equation

In quantum mechanics, the state in the instant t of a system is described by an element 日本語 (t) {displaystyle scriptstyle leftALESPsi (t)rightrangle } of Hilbert's complex space — using Paul Dirac's bra-ket notation. The probability of results of all possible measures of a system can be obtained from 日本語 (t) {displaystyle scriptstyle leftALESPsi (t)rightrangle }. Temporary evolution 日本語 (t) {displaystyle scriptstyle leftALESPsi (t)rightrangle } is described by the Schrödinger equation:

H^ ^ 日本語 (t) =i ddt日本語 (t) =p→ → ^ ^ 22m日本語 (t) +V(r→ → ^ ^ ,t)日本語 (t) {displaystyle mathbf {hat {H} left englishPsi (t)rightrangle =ihbar {d over dt}left

where

• i{displaystyle i,}: is the imaginary unit;
• {displaystyle hbar ,}: is the standardized Planck constant (h/2π);
• H^ ^ {displaystyle {hat {H},}: is the hamiltonian, dependent on the overall time, the observable corresponds to the total energy of the system;
• r→ → ^ ^ {displaystyle {hat {vec {mathbf {r}}}}{,}: is the observable position;
• p→ → ^ ^ {displaystyle {hat {vec {mathbf {p}}}}{,}: is the observable impulse.
• V{displaystyle V,}: is the potential energy

As with the force in Newton's second law, its exact form does not give the Schrödinger equation, and has to be determined independently, from the physical properties of the quantum system.

It should be noted that, contrary to Maxwell's equations that describe the evolution of electromagnetic waves, Schrödinger's equation is non-relativistic. Also note that this equation is not proved: it is a postulate. It is assumed to be correct after Davisson and Germer experimentally confirmed Louis de Broglie's hypothesis.

For more information on the role of operators in quantum mechanics, see the mathematical formulation of quantum mechanics.

Limitations of the equation

• Schrödinger's equation is a non-relativist equation that can only describe particles whose linear moment is small compared to the resting energy divided by the speed of light (if this condition is not fulfilled, it must be applied to a relativistic equation such as Dirac's equation or Klein-Gordon's).
• In addition, Schrödinger's equation does not incorporate the sphin of the particles properly. Pauli slightly generalized Schrödinger's equation by introducing in it terms that correctly predicted the effect of the sphin; the resulting equation is Pauli's equation.
• Later, Paul Dirac, provided the now-called Dirac equation that not only incorporated the splint for splinters 1/2, but introduced the relativistic effects.

Solving the equation

Schrödinger's equation, being a vector equation, can be rewritten in an equivalent way in a particular basis of state space. If chosen for example the generalized basis 日本語r→ → {displaystyle leftATA{vec {r}}rightrangle } for the representation of position defined by:

r→ → ^ ^ 日本語r→ → =r→ → 日本語r→ → {displaystyle {hat {vec {mathbf {r}}}}}{left ultimate{vec {r}rightrangle ={vec {r}}}{left

Then the wave function (t,r→ → )≡ ≡ r→ → 日本語 (t) {displaystyle scriptstyle Psi (t,{vec {r}})equiv leftlangle {vec {r}}right responsibleleft.Psi (t)rightrangle ,} meets the following equation:

i ▪ ▪ (t,r→ → )▪ ▪ t=− − 22m► ► → → 2 (t,r→ → )+V(r→ → ,t) (t,r→ → ){displaystyle ihbar {partial Psi (t,{vec {r}}) over partial} t=-{hbar ^{2} over 2m}{overrightarrow {nabla }}}{2}Psi (t,{vec {r}{vec {r}{,t}{p)}{

Where ► ► → → 2{displaystyle {overrightarrow {nabla }}^{2},} It's the laplatian. In this way it is seen that Schrödinger's equation is an equation in partial derivatives in which linear operators intervene, which allows to write the generic solution as a sum of particular solutions. The equation is in the vast majority of cases, too complicated to admit an analytical solution so that its resolution is made approximately and/or numerically.

Note that the wave function defined thus, for linked states can always be interpreted as an element of the complex and separable Hilbert space L2(R3){displaystyle scriptstyle L^{2}(mathbb {R} ^{3}}}, although for collision states or not linked it is necessary to go to Hilbert spaces equipped for rigorous treatment.

Search for eigenstates

The operators that appear in the Schrödinger equation are linear; from which it follows that every linear combination of solutions is a solution of the equation. This leads to favor the search for solutions that have great theoretical and practical interest: knowing the states that are typical of the Hamiltonian operator. These states, called stationary states, are the solutions of the equation of states and eigenvalues,

H^ ^ 日本語φ φ n =En日本語φ φ n {displaystyle {hat {H}} meantvarphi _{n}rangle =E_{n}{n}{n}{n}{n}{rangle }

habitually Schrödinger equation independent of time. The state itself 日本語φ φ n {displaystyle Čvarphi _{n}rangle } is associated with its own value En{displaystyle E_{n}}, real scale that corresponds to the energy of the particle in that state.

The energy values can be discrete as the solutions bound to a potential well (for example level of the hydrogen atom); resulting in a quantization of energy levels. These can also correspond to a continuous spectrum such as the free solutions of a potential well (for example, an electron that has enough energy to move infinitely away from the nucleus of a hydrogen atom).

Often you get numerous states 日本語φ φ n {displaystyle Čvarphi _{n}rangle } They correspond to the same value of energy: we speak then of degenerate energy levels.

In general, the determination of each of the states of the Hamiltonian, 日本語φ φ n {displaystyle Čvarphi _{n}rangle }, and of the associated energy, gives the corresponding stationary state, solution of the Schrödinger equation:

日本語END END n(t) =日本語φ φ n Exp (− − iEnt ){displaystyle Δpsi _{n}(t)rangle ,=, ultimatevarphi _{n}rangle ,exp left({frac {-iE_{n}t}{hbar }}{right)}

A solution of the Schrödinger equation can then generally be written as a linear combination of such states:

日本語END END (t) =␡ ␡ n␡ ␡ icn,i日本語φ φ n,i Exp (− − iEnt ){displaystyle Δpsi (t)rangle ,=,sum _{n}sum _{i}c_{n,i}{n,i}{n,i}rangle exp left({frac {-iE_{n}t}{hbar }}{right)}}

According to the postulates of quantum mechanics,

• the complex climbing cn,i{displaystyle c_{n,i} is the extent of the state 日本語END END (t) {displaystyle Δpsi (t)rangle } about the state 日本語φ φ n,i {displaystyle Δvarphi _{n,i}rangle };
• the real ・ ・ i日本語cn,i日本語2_ is the probability (in the case of a discreet spectrum) of finding the energy En{displaystyle E_{n}} while making a measure of energy on the system.

Rarity of an exact analytical solution

The search for eigenstates of the Hamiltonian is generally complex. Even in the analytically solvable case of the hydrogen atom, it is only rigorously solvable in a simple way if the coupling with the electromagnetic field that allows the passage to excited states, solutions of the Schrödinger equation of the atom, is ruled out from the fundamental level.

Some simple models, although not entirely consistent with reality, can be solved analytically and are very useful. These solutions serve to better understand the nature of quantum phenomena, and are sometimes a reasonable approximation to the behavior of more complex systems (in statistical mechanics molecular vibrations are approximated as harmonic oscillators). Examples of models:

• The free particle (potential null);
• The particle in a box
• A particle beam affecting a potential barrier
• The particle in a ring
• The particle in a spherical symmetry potential
• The quantum harmonic oscillator (quarter potential)
• The hydrogen atom (potential of spherical symmetry)
• The particle in a single-dimensional network (period potential)

In other cases, approximation techniques must be used:

• The disturbing theory gives analytical expressions in the form of asymptotic developments around a problem without-perturbations that is exactly resoluble.
• Numerical analysis allows to explore inaccessible cases to the theory of disturbances.
• The Variation Method
• Hartree-Fock solutions
• The quantum methods of Monte Carlo

Classical limit of the Schrödinger equation

Initially the Schrödinger equation was considered simply as the equation of movement of a material field that spread in the form of wave. In fact it can be seen that in the classic limit, when → → 0{displaystyle hbar to 0} Schrödinger's equation is reduced to the classic equation of movement in terms of action or equation of Hamilton-Jacobi. To see this, we will work with the typical wave function that satisfies the Schrödinger equation dependent on the time it has the form:

END END (x,t)=eiS(x,t)/ {displaystyle psi (x,t)=e^{iS(x,t)/hbar }}

Where S(x,t)/ {displaystyle S(x,t)/hbar } is the phase of the wave if this solution is replaced in the time-dependent Schrödinger equation, after reordering the terms conveniently, it comes to:

(4)▪ ▪ S▪ ▪ t+12m[chuckles](▪ ▪ S▪ ▪ x)2+(▪ ▪ S▪ ▪ and)2+(▪ ▪ S▪ ▪ z)2]+V(x)=i 2mΔ Δ S{displaystyle {frac {partial S}{partial t}} +{frac {1}{2m}}left({frac {partial S}{partial S}{partial }{ right}{2⁄2⁄2}{left({partal S}{partial}{right}{2⁄2⁄2}{left

If you take the limit → → 0{displaystyle hbar to 0} the second member disappears and we have that the phase of the wave function coincides with the magnitude of action and this magnitude can be taken as real. Equally since the magnitude of action is proportional to the mass of a particle S=msm{displaystyle S=ms_{m},} It can be seen that for large mass particles the second member is much smaller than the first:

(5)▪ ▪ sm▪ ▪ t+12 ► ► → → sm 2+V(x)=limm→ → ∞ ∞ i 2mΔ Δ sm=0{displaystyle {frac {partial s_{m}}{partial t}}}{+{frac {1}{2}}leftVert {vec {nabla }}}s_{mrightVert ^{2}+V(x)=lim _{mto infty }{frac}{itahbar}{

And therefore for macroscopic particles, given the smallness of Planck's constant, the quantum effects summarized on the right hand side cancel out, which explains why quantum effects are only appreciable at subatomic scales.

According to the correspondence principle, classical particles of great mass, compared to the quantum scale, are localized particles describable by a highly localized wave packet moving through space. The wavelength of the waves that made up said material packet are around the de Broglie length for the particle, and the group velocity of the packet coincides with the velocity of the particle's motion, which reconciles the corpuscular nature observed in certain experiments with the wave nature observed for subatomic particles.

Matrix formulation

There is a matrix formulation of quantum mechanics, in this formulation there is an equation whose form is essentially the same as that of the classical equations of motion, said equation is:

(6)dA^ ^ dt=▪ ▪ A^ ^ ▪ ▪ t− − i [chuckles]A^ ^ ,H^ ^ ]{displaystyle {frac {{text{d}}{hat {A}}}}{{{{{text{d}}}t}{,=frac {partial {A}}}}}{partial t}}}-{frac {i}{hslash }}[{hat {A}, {,{hat {H}}}}}}}{,{{{{{

From this equation it is possible to deduce Newton's second law, solving for the operator p^ ^ {displaystyle {hat {p}}}. Indeed it has

(7)dp^ ^ dt=− − i [chuckles]p^ ^ ,H^ ^ ]{displaystyle {frac {{text{d}}{hat {p}}}{{{{{text{d}}}t}{,=,-{frac {i}{hslash }}}}}{{{{hat {p}}},{,{hat {h}}}}}}}}}}}

by evaluating the commutator it follows

(8)dp^ ^ dt=− − i (p^ ^ H^ ^ − − H^ ^ p^ ^ )=− − i (p^ ^ 32m+p^ ^ V− − p^ ^ 32m− − Vp^ ^ )=− − i (p^ ^ V− − Vp^ ^ ){cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFFFF}{cHFFFFFFFF}{cH}{cHFFFF}{cHFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFFFFFFFF}{cH}{cH}{cH}{cHFFFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFFFFFFFFFFFF}{cH}{cHFFFFFF}{cHFFFFFFFFFFFFFFFFFFFFFFFFFF}{cHFFFFFFFFFFFFFFFFFF}{cHFFFFFFFF}{cHFF}{cHFF}{cHFFFFFFFF}{cHFFFFFF}{cHFFFFFFFFFFFF}{c

It's not hard to prove that Vp^ ^ =0{displaystyle V{hat {p}}=0} and therefore it is obtained:

(9)dp^ ^ dt=− − i p^ ^ V=− − ► ► V{displaystyle {frac {{text{d}}{hat {p}}}{{{{{text{d}}}t},=,-{frac {i}{hslash }}}{hat {p}}V,=,-{boldsymbol {nab }V}}

where it has been used p^ ^ =− − i ► ► {displaystyle {hat {p}}=-ihslash {boldsymbol {nabla}}}}. This result is analogous to that of classical mechanics, for a similar equation involving the Poisson brackets, moreover, this equation is precisely the Newtonian formulation of mechanics.