Schroedinger equation
physical treatment
the Schroedinger equationis the principal equation of nonrelativistic quantum mechanics. It describes the temporal development of the condition of a quantum system. The Schroedinger equation was set up to 1926 of Erwin Schroedinger first as wave equation, represented afterwards by Werner Heisenberg equivalent as operator equation. As „motion equation of quantum mechanics “formsit this very day the foundation for nearly all practical applications of quantum mechanics.
A simpler understanding of the theory forms the thought experiment of Schroedinger cat.
the equation
the Schroedinger equation reads with absence of a magnetic field for an individual particle (aboutan elementary particle or an atom) in the Potenzial V (for example the gravitation Potenzial), whose condition is described by (scalar, often like here by the Greek letter psi expressed ) the wave function ψ:
- <math> \ mathrm {i} \ cdot \ hbar \ cdot \ frac {\ partial} {\ partial t} \ psi (\ mathbf {r}, t) \; = \; - \ frac {\ hbar^2} {2m} \ cdot \ nabla^2 \ psi (\ mathbf {r}, t) + V (\ mathbf {r}, t) \ cdot \ psi (\ mathbf {r}, t)< /math> .
Is
- m the mass of the particle,
- <math> \ mathbf {r} \ in \ R^3< /math> the place as well as <math> \ nabla< /math> the partial derivative after the place (Del),
- t the time as well as <math> \ frac {\ partial} {\ partial t}< /math> the partial derivative after the time,
- <math> \ mathrm {i}< /math> the imaginary unit of the complex numbers,
- < math> \ hbar= {h \ more over2 \ pi}< /math> (speak „hectar crosswise “) with the Planck quantum of action <math> h< /math> (a natural constant, which does not occur in „classical “physics).
With a free particle V (r, t) applies = 0.
By means of the Hamilton operator H the Schroedinger equation leaves itself psi just as well
- < cdot> as math \ mathrm {i} \ hbar \ to t \ frac {\ partial} {\ partial} \ (\ mathbf {r}, t)= \ H has \ psi (\ mathbf {r}, t)< /math>
with
- <math> \ H = \ cdot frac {1} {2m} \ (\ mathrm {i} \ hbar \ nabla) ^2 + V (\ mathbf {r}, t)< /math>
write.
derivation
the Schroedinger equation develops for E = \ according to the correspondence principle from the classical principle of conservation of energy (total energy corresponds to the sum of kinetic and
- <potenzieller> energy) math frac {\ mathbf {p} ^2} {2m} + V (\ mathbf {r}, t)< /math>
by replacing the classical sizes of energy, impulse and place by the appropriate quantum-mechanical operators (in the local representation)
- <math> \ begin {} E & \ rightarrow& \ E has matrix &=& \ mathrm {i} \ hbar \ cdot \ frac {\ partial} {\ partial t} \ \ \ mathbf {p} & \ rightarrow& \ mathbf {\ has p} &=& - \ mathrm {i} \ hbar \ cdot \ nabla \ \ \ mathbf {r} & \ rightarrow& \ mathbf {\ has r} &=& \ mathbf {r} \ ends {to matrix}< /math>
andfollowing applying to the unknown function ψ :
- <math> 0 = \ (\ E - \ frac {\ mathbf {\ p has} ^2} {2m} - has V (\ mathbf {\ r has}, t) \ right) \ psi (\ mathbf {r}, t) left< /math> .
Another beginning for the derivation of the Schroedinger equation goes as follows: Similar to to even light waves particles can in extreme cases as de Broglie wave are understood:
- <math> \ psi (\ mathbf {r}, t) = \ operator name {const} \ cdot \ exp \ left (\ frac {- \ mathrm {i}} {\ hbar} \ cdot (E \ cdot t - \ mathbf {p} \ cdot \ mathbf {r}) \ right)< /math> ;
for V (r, t) = 0 this wave of the Schroedinger equation is sufficient. In both beginnings the nature of the wave function remains for the time being unknown ψ; their square amount <math>|\ psi|^2< /math> leaves itself however physical asAmplitude of the Wahrscheinlichkeitsdichte of the particle understand.
Remark: Actually the Schroedinger equation cannot be deduced from classical physics, since quantum mechanics is a Verallgemeinerung of classical physics. Schroedinger has rather from at his time the already well-known quantum-mechanical phenomena ofParticle (with consideration of certain physical principles) these phenomena describing equation designs.
characteristics
ψ is always a komplexwertige function (if the left expression is not identically 0). Because the Schroedinger equation is relative ψ linear and homogeneous, is for a given solution ψ also each scalar multiple <math> k \ cdot \ psi< /math> with a complex constant <math> k \ in \ Bbb C< /math> a solution. Due to this ambiguity it is meaningful and usual, only standardized solutions in the sense of the standardisation condition
- <math> \ int_ {} ^ {} |\ psi|^2 (\ mathbf {r}, t) \; \ mathrm {D} ^3r = 1< /math>
to regard. Howevereach solution of a Schroedinger equation is not standardizable. If existence is, this standardized solution up to a phase factor of the form <math> \ exp (\ mathrm {i} \ cdot K)< /math> for a real K, which is physically insignificant however, clearly intends. For a standardized solution becomes <math>|\ psi|^2 (\ mathbf {r}, t) =\ psi^* \ \ psi /math< cdot> as Wahrscheinlichkeits-Dichte for the fact that itself the particle at the place <math> \ mathbf {r}< /math> at the time t finds, interpreted. Thus also the selected standardisation condition receives its meaningful interpretation.
Generally it applies that not only multiple but arbitrary linear combinations from atomic orbitals to the same energy value againa permissible solution for this atom give. This follows from the linearity of the Hamilton operator. <math> H (A \ psi+b \ chi) =aH \ psi+bH \ chi=aE \ psi+bE \ chi=E (A \ psi+b \ chi)< /math> The enstehenden orbital hybrid orbital are called and to play an important role in the valence bond theory. In order to find molecule orbital, one proceeds similarly. Becomebut the wave functions of different atoms linear combines. And the developing thing molecule-orbitally is only one approximation.
one has to solve
solution of the Schroedinger equation in case of a really time-dependent Hamilton operator H = H (r, t) a task of initial value: One knows thoseMass m of the particle, the Potenzial V ( r, t), put on from the outside, the initial condition ψ (r, 0) = given as well as the boundary conditions too ψ for t > , then one receives the wave function to 0 with the help of the Schroedinger equation as solution ψ (r, t) for thatregarded area for all times t > 0.
In case of a time-independent Hamilton operator H = H (r) and firm edges (thus in particular time-independent Potenziale V = V (r)) the wave function places ψ = ψ (r, t) against it a so-called stationary condition (oran overlay from these), and one has to solve then a task of boundary value: One brings the beginning of the Separation of the variables <math> \ to psi (\ mathbf {r}, t) = \ psi (\ mathbf {r}) \ cdot \ chi (t)< /math> into the Schroedinger equation, then one receives with the real constant of E the equation
- < math> \ to psi (\ mathbf {r}, t)= \ psi (\ mathbf {r}) \ cdot \ exp ({- \ frac {\ mathrm {i}} {\ hbar} E t})< /math>
and the far time-independent Schroedinger equation
- <math> \ H (\ mathbf {r}) has \ psi (\ mathbf {r}) = E \ cdot \ psi (\ mathbf {r})< /math> .
Together with the homogeneous and linear boundary conditions ψ (r) the time-independent Schroedinger equation on forms a so-called Eigenwertaufgabe, with the energy eigenvalues E and proper functions ψ (r) toodetermine are. A simple example such a Eigenwertaufgabe forms an electron without Potenzial in a box.
From in such a way found wave function all physical characteristics of the particle result. For example the classical value becomes <math> \ mathbf {r} (t)< /math> by the middle place of the particleat present t, i.e.
- <math> \ mathbf {r} (t) \ rightarrow \ langle \ mathbf {\ cdot r} \ rangle (t) = \ int_ {} ^ {} \ mathbf {r} \|\ psi|^2 (\ mathbf {r}, t) \ mathrm {D} ^3r< /math>
replaced, while for example the classical value of the impulse is replaced by the following average value:
- <math> \ mathbf {p} (t) \ rightarrow \ langle \ mathbf {\ has p} \ rangle (t) = \ int_ {} ^ {} \ psi^* (\ mathbf {r}, t) (- \ mathrm {i} \ hbar \ cdot \ nabla) \ psi (\ mathbf {r}, t) \ mathrm {D} ^3r< /math> .
In principle (!) becomes anyclassical measured variable f (r, p, t) by an averaging of the associated operator over the area, in which the particle can be, replaces:
- <math> f (\ mathbf {r} (t), \ mathbf {p} (t), t) \ rightarrow \ langle \ has f \ rangle (t) = \ int_ {} ^ {} \ psi^* (\ mathbf {r}, t) f (\ mathbf {\ r has}, \ mathbf {\ has p}, t) \ psi (\ mathbf {r}, t) \ mathrm {D} ^3r< /math> .
One calls the expression <f> the expectancy value of f.The expectancy value of the energy is equal to< H>.
views and explanations
the Schroedinger equation is contrary to the classical force equations a partial differential equation. While a classical particle by an accurate course <math> \ mathbf {r} (t)< /math> , becomes those is certainDynamics of the quantum-mechanical particle by the quantum-mechanical field ψ described. The classical Newton's motion equation
- <math> m \ cdot \ frac {\ mathrm {D} ^2 \ mathbf {r}} {\ mathrm {D} t^2} (t) = - \ nabla V (\ mathbf {r} (t), t)< /math>
thus in quantum mechanics by the Schroedinger equation one replaces. In quantum mechanics therefore an accurate place of residence (generally) is not definably; descriptive one, the particle says is” smeared “over the area. In the border line that those is sufficient small width of the wave packet, it can be deduced with the help of the Schroedinger equation the Newton's equation.
The Schroedinger equation indicated above is that in the so-calledLocal representation. In the form independent of a certain basis (like the place) reads the Schroedinger equation with “ket”|t>:
- <math> \ mathrm {i} \ hbar \ cdot \ frac {\ partial} {\ partial t}|t \ rangle = \ frac {\ mathbf {\ has p} ^2} {2m}|t \ rangle + V (\ mathbf {\ r has}, t)|t \ rangle< /math>,
whereby here math \ <mathbf> {\ p has the operators}< /math> and <math> \ mathbf {\ r has}< /math> likewise as basis independentto regard are. Both equations are equivalent.
In the Schroedinger equation the wave function and the operators in the so-called Schroedinger picture occur. In the case of use of the wave function and the operators in the Heisenberg picture the Heisenberg equation results. Both equations are physically equivalent.
The Schroedinger equationis on the one hand deterministic; that is, the physics of the particle is exactly certain; in particular the Schroedinger equation is not” somehow incomplete “. On the other hand their solution is ψ however a statistic size; ψ only a statement about the whole of all homogeneous experimental assemblies therefore makes.Generally this has as a consequence: If one in each case accomplishes at two physically identical systems the same measurement (about those of the place of the particle at the same time), then both measured values (contrary to classical physics) can fail differently.
The Schroedinger equation containswith the Planck quantum of action h a size, which was found before by Max Planck for light quanta. It was consulted by the Schroedinger equation also for the description by particles such as electrons.
With the formulation of the Schroedinger equation the contradictory construction became of the Bohr atom model overcome. That is, with the Schroedinger equation one was for the first time able to compute the hydrogen atom in good approximation without having to hurt thereby the laws of electrodynamics.
The Schroedinger equation has however the following fundamental lack: Itknows nothing from the spin moment (spin) of the particle; in addition it is not lorentzinvariant (but “only” galilei invariantly). A relevant advancement of the Schroedinger equation represents for example for electrons the relativistic Dirac equation , which is to be handled however also substantially more difficult.
Hamilton operator for charged particles in the electromagnetic field
of case the particle an electrical charge possesses (for example an electron or a proton is), then the Hamilton operator (in the local representation) generalizes itself too math \ has H with presence
- <> of an outside electromagnetic field= \ frac {1} {2m} \ cdot \ left (- \ mathrm {i} \ hbar \ nabla - \ frac {q} {C} \ cdot \ mathbf {A} (\ mathbf {r}, t) \ right) ^2 + q \ cdot \ Phi (\ mathbf {r}, t) + V (\ mathbf {r}, t)< /math> ,
whereby here q the electrical charge of the particle (q = - e with electrons), C the speed of light in the vacuum, <math> \ mathbf {A}< /math> the so-called Vektorpotenzial and Φ the scalar Potenzial designate. Thoseso resulting in Schroedinger equation steps itself to the place of the classical Lorentzgleichung. For the outside electrical field <math> \ mathfrak {E}< /math> and the magnetic field <math> \ mathfrak {H}< /math> exists the following relations:
- <math> \ mathfrak {E} (\ mathbf {r}, t) = - \ nabla \ Phi (\ mathbf {r}, t) - \ frac {1} {C} \ cdot \ frac {\ partial \ mathbf {A}} {\ partial t} (\ mathbf {r}, t)< /math>
- <math> \ mathfrak {H} (\ mathbf {r}, t) = \ nabla \ times \ mathbf {A} (\ mathbf {r}, t)< /math> .
Thusthe outside electromagnetic field works (<math> \ mathfrak {E}, \ mathfrak {B}< /math>) (over the Schroedinger equation) on the field ψ.
Turned around in principle the wave function affects ψ the outside electromagnetic field in the following way: Out ψ and q compute themselves the electrical current density and the charge densitythe particle. It produces such its own an electromagnetic field, which retroacts on all other charges and rivers, whatever cause the outside field.
The Schroedinger equation does not consider the reciprocal effect of the spin moment (spin) of the particle with the outside magnetic field. For example(substantially more complicated) the Pauli equation is to be used for an electron with presence of an outside magnetic field, if the reciprocal effect between spin and magnetic field is not negligible.
lying rank density of the Schroedinger equation
the lying rank density of the Schroedinger equation reads
- < math> \ mathcal {L} (\ psi, \ vec {\ nabla} \ psi, \ DOT {\ psi}) = i \ hbar \,\ psi^ {*} \ DOT {\ psi} - \ frac {h^2} {2m} \ vec {\ nabla} \ psi^ {*} \ cdot \ vec {\ nabla} \ psi - V (\ vec {r}) \, \ psi^ {*} \ psi< /math>
applications
the Schroedinger equation can be solved for some simple Potenziale accurately, e.g.:
- the particle in the box
- the particle in the spherically symmetric Potenzial (hydrogen atom)
- the Potenzialbarriere (tunnel effect ) that
- results in harmonious Potenzial
already with the H 2 + - ion is no longer possible an accurate solution. Therefore those must for more than two particles, for example with multi-electron systems,Schroedinger equation to be simplified or approximately solved. A possible simplification is the boron burl home he approximation, in addition, perturbation theory can supply good approximations. An analytic solution is still impossible, like e.g. despite the simplification. with most atoms and all molecules, then iterative approximation methods must be used. Within the range of theoretical chemistrythe Hartree Fock method is for this often used.
an analogy of the linear Schroedinger equation to the wave equation
Erwin Schroedinger accepted 1926, supported by the investigations de Broglies that all quantum objects have „Wellennatur “. Mathematically one would have it therefore by a wave function<math> \ psi (\ vec x, t)< /math> (Psi function) clearly to describe can. In particular a free particle (math V ( <x>) applies for =0 /math here<)> should after de Broglie by an even wave with rotative frequency <math> \ omega = e \ hbar< /math> and Wellenzahl< math> \ vec k = \ vec p \ hbar< /math> are described. Such a wave has thoseShape
- < math> \ psi (x, t) = \ mathrm {e} ^ {\ mathrm {i} (\ vec k \ cdot \ vec x - \ omega t)}</math>.
Now psi (\ vec x,
- < t>) applies = \ mathrm {for i} \ vec k \ psi (\ vec x, t) /math to the wave indicated above math \ vec \ nabla< \>
- <math> \ frac {\ partial} {\ partial t} \ psi = - \ mathrm {i} \ omega \ psi (\ vec x, t)< /math>
Together with the De-Broglie-formulas for energy and impulseresult thus the operator equations
- < math> - \ mathrm {i} \ hbar \ vec \ nabla \ psi (\ vec x, t) = \ vec p \ psi (\ vec x, t)< /math>
- <math> \ mathrm {i} \ hbar \ frac {\ partial} {\ partial t} \ psi (\ vec x, t) = E \ psi (x, t)< /math>
In addition the Newton's energy impulse relationship must apply to free particles:
- <math> E = \ frac {\ vec p^2} {2m}< /math>
Multiplies one both sides by <math> \ psi (\ vec x,t)< /math>, then can be used the two operator equations for energy and impulse, and one receives:
- <math> \ mathrm {i} \ hbar \ frac {\ partial} {\ partial t} \ psi (\ vec x, t) = - \ frac {\ vec \ nabla^2} {2m} \ psi (\ vec x, t)< /math>
That is the Schroedinger equation for a free particle.
One assumes that the operator equations also for particlesin the potential apply, then one receives from the full Newton's energy impulse relationship:
- <math> E = \ frac {\ vec p^2} {2m} + V (x)< /math>
in the same way the full Schroedinger equation.
A system is temporally not variable (electron in the box, hydrogen atom…), then one knows the equation stronglysimplify. This succeeds by means of the so-called. Separate ion beginning (Separation of the time):
- <math> \ psi (\ vec x, t) = \ psi (\ vec x) \ cdot \ phi (t)< /math>
If one inserts this function into the Schroedinger equation, then the time derivative affects only <math> \ phi< /math> and the time-independent Hamilton operator only on <math> \ psi< /math>. One divides those, One keeps hbar therefore math \ frac {\ mathrm
- < {>i to Schroedinger equation on both sides by the wave function} \ \ partial \ phi (t)/\ partial t} {\ phi (t)} = \ frac {\ H has \ psi (\ vec x)}{\ psi (\ vec x)}</math>
There the left side not of <math> \ vec x< /math> separate only from <math> t< /math> , the right side depends however only on <math> \ vec x< /math> and notby <math> t< /math>, both sides must be constant, so that the equation can be fulfilled. This constant is the energy E of the wave function. Thus one keeps the two equations
- < hbar> math \ mathrm {to i} \ \ frac {\ partial} {\ partial t} \ phi (t) = E \ phi (t)< /math>
- <math> \ H has \ psi (\ vec x) = E \ psi (\ vec x)< /math>
The first equationthe solution clear up to a standardisation constant has
- < math> \ phi (t) = \ mathrm {e} ^ {\ mathrm {- i} Et/\ hbar}< /math>
The second equation is the time-independent or stationary Schroedinger equation. It is a differential equation 2. Order and can be applied to many quantum-mechanical problems.
Hamilton operator forMolecules
- of the kinetic energy of the electrons <math> T_e< /math>
- the kinetic energy of the atomic nuclei <math> T_k< /math>
- the potential energy of the reciprocal effects between the electrons <math> V_ {ee}< /math>
- the potential energy of the reciprocal effects between the atomic nuclei <math> V_ {kk}< /math>
- the potential energy of the reciprocal effects between the electrons andAtomic nuclei <math> V_ {ek}< /math>
It is usual, the Hamilton operator not in SI-UNITs to write but in so-called atomic units since this saves the following advantages:
- Since natural constants do not emerge any longer explicitly, the results are in atomic units more simply in addition-write and independently ofthe accuracy of the involved natural constants. The sizes computed in atomic units can be reckoned back nevertheless simply in SI-UNITs.
- Numeric solution procedures of the Schroedinger equation behave more pleasantly, since the numbers which can be processed are substantially more near with the number of 1, than this inSI-UNITs the case is.
The Hamilton operator surrenders too
- < math> H = T_e + T_k + V_ {ee} + V_ {kk} + V_ {ek}< /math>
with
- <math> T_e = - \ frac {1} {2} \ sum_ {i=1} ^N \ Delta_i </math>,
- <math> T_k = - \ frac {1} {2} \ sum_ {\ mu=1} ^N \ frac {1} {m_ \ mu} \ Delta_ \ mu </math>,
- <math> V_ {ee} = \ sum_ {i=1} ^ {N-1} \ sum_ {j> i} ^N \ frac {1} {r_ {ij}}< /math>,
- <math> V_ {kk} = \ sum_ {\ mu=1} ^ {M-1} \ sum_ {\ nu> \ mu} ^M \ frac {Z_ {\ mu} Z_ {\ nu}} {R_ {\ mu \ nu}}< /math>,
- <math> V_ {ek} = - \ sum_ {i=1} ^N \ sum_ {\ mu=1} ^M \ frac {Z_ {\ mu}} {R_ {i \ mu}}< /math>.
Here math <\> delta is </math> the Laplace operator, <math> i< /math> and <math> j< /math> the indices over the electrons, <math> \ mu< /math> and/or. <math> \ nu< /math> the indices over the atomic nuclei, <math> r_ {ij}< /math> the distance between ithand that j-ten electron, <math> R_ {\ mu \ nu}< /math> the distance between <math> \ mu< /math> - ten and <math> \ nu< /math> - atomic nucleus and math <R_> ten {i \ mu}< /math> the distance between <math> the i< /math> - electron and math <\> mu /math< ten> - atomic nucleus, math <Z_> ten {\ mu}< /math> the nuclear charge number <math> \ mu< of /math> - ten atomic nucleus.
The time-independent Schroedinger equation results then too <math> H\ Psi = E \ psi </math>, whereby however in practice the total Schroedinger equation is divided with the help of the boron burl home he approximations in an electronic Schroedinger equation (with firm core coordinates) and a core Schroedinger equation. The solution of the core Schroedinger equation sets thereby the solution of the electronic Schroedinger equation forall (relevant) core geometry ahead, since the electronic energy is received there as function of core geometry. The electronic Schroedinger equation results formally through sets from <math> T_k = 0< /math>.
mathematical treatment
outside of physics enjoys the Schroedinger equation in mathematicsa high measure of interest. A large number of mathematicians concerns itself also up-to-date with the investigation of existence - and clarity questions, qualitative investigations of the characteristics of special solutions of the equations (e.g. examining Solitonen) and that numeric solution of the equations.
form of the equation
nature is not for a mathematician necessarily of such importance as for a physicist. Therefore the constants are omitted, which way of writing partly abstracted. The Schroedinger equation as Cauchyproblem hasthe following shape:
- <math> i \ frac {\ partial u} {\ partial t} - \ delta u = f (u), \ quad u (0) =u_0, </math>
whereby <math> \ Delta </math> the Laplaceoperator is.
fitting areas
the treatment of the Schroedinger equation are selected uniformly the Sobolevräume. We designate here for <math> \ Omega \ subseteq \ R^n</math>
- <math> H^s (\ omega) \; = \; \ {f:\Omega\rightarrow\R \,|\, D^ \ alpha f \ in L^2 \, \ forall |\ alpha|\ leq s \} </math> for integral s and
- <math> H^s (\ omega) \; = \; \ {f \ in S'(\ omega) \,|\, F^ {- 1} ((1+|\ xi|^2) ^s \ has {f}) \ in L^2 \}, </math> for <math> s \ in \ R </math>,
whereby <math> S' </math> that Dual area of the black functions, <math> \ {u has} </math> the fourier transformation as well as <math> F^ {- 1} </math> the inverse transform designates. It has itself e.g. for <math> s= \ frac {1} {2} </math> the linguistic usage a half derivative interspersed. Also one can define quite negative exponents: A function in<math> H^ {- 2} </math> here is a function (or rather distribution) is, those by twice deriving of one <math> L^2< /math> - function develops.
characteristics of solutions
preservation <math> of the H^s< /math> - standards
- < math> ||u (t, \ cdot)||_ {H^s} \; = \; ||u_0||_ {H^s} </math>
To see simpleby fourier transformation of the Semigruppe.
Propagation of information with infinite speed
is <math> u_0 \ in S'(\ R^n) </math>. It exists <math> \ varphi_n \ in S'(\ R^n) </math> with <math> \ varphi_n \ longrightarrow u_0< /math>. Is now <math> u_n </math> Solution of the equation
- < math>i \ partial_t u_n - \ delta u_n \, = \, 0, \ quad u_n (\ cdot, 0) = \ varphi_n, </math>
then applies
- <math> u_n \ longrightarrow u </math>
dispersion
it applies
- <math> ||u (t, \ cdot)||_ {L^ \ infty} \, \ leq \, \ frac {C} {|t|^ {n/2}} ||u_0||_ {L^1} \ quad \ forall u_0 \ in L^1 </math>
