In quantum mechanics, the
Contents 1 Equation 1.1 Time-dependent equation 1.2 Time-independent equation 2 Derivation 3 Implications 3.1 Total, kinetic, and potential energy
3.2 Quantization
3.3 Measurement and uncertainty
3.4
4 Interpretation of the wave function 5 Historical background and development 6 The wave equation for particles 6.1 Consistency with energy conservation 6.2 Linearity 6.3 Consistency with the De Broglie relations 6.4 Wave and particle motion 7 Nonrelativistic quantum mechanics 7.1
7.2.1 Free particle 7.2.2 Constant potential 7.2.3 Harmonic oscillator 7.3 Three-dimensional examples 7.3.1
7.4
8 Solution methods 9 Properties 9.1 Linearity
9.2
10 Relativistic quantum mechanics
11
Equation[edit]
Time-dependent equation[edit]
The form of the
A wave function that satisfies the nonrelativistic Schrödinger equation with V = 0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here. Time-dependent
i ℏ ∂ ∂ t Ψ ( r , t ) ⟩ = H ^ Ψ ( r , t ) ⟩ displaystyle ihbar frac partial partial t vert Psi (mathbf r ,t)rangle = hat H vert Psi (mathbf r ,t)rangle where i is the imaginary unit, ħ is the reduced
ℏ = h 2 π displaystyle hbar = frac h 2pi , 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, r and t are the position vector and time respectively, and Ĥ is the Hamiltonian operator (which characterises the total energy of the system under consideration). Each of these three rows is a wave function which satisfies the
time-dependent
The most famous example is the nonrelativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field; see the Pauli equation):[6] Time-dependent
i ℏ ∂ ∂ t Ψ ( r , t ) = [ − ℏ 2 2 μ ∇ 2 + V ( r , t ) ] Ψ ( r , t ) displaystyle ihbar frac partial partial t Psi (mathbf r ,t)=left[ frac -hbar ^ 2 2mu nabla ^ 2 +V(mathbf r ,t)right]Psi (mathbf r ,t) where μ is the particle's "reduced mass", V is its potential energy,
∇2 is the
Time-independent
H ^ Ψ ⟩ = E Ψ ⟩ displaystyle operatorname hat H vert Psi rangle =Evert Psi rangle where E is a constant equal to the total energy of the system. This is only used when the Hamiltonian itself is not dependent on time explicitly. However, even in this case the total wave function still has a time dependency. In words, the equation states: When the Hamiltonian operator acts on a certain wave function Ψ, and the result is proportional to the same wave function Ψ, then Ψ is a stationary state, and the proportionality constant, E, is the energy of the state Ψ. In linear algebra terminology, this equation is an eigenvalue equation
and in this sense the wave function is an eigenfunction of the
Hamiltonian operator.
As before, the most common manifestation is the nonrelativistic
Time-independent
[ − ℏ 2 2 μ ∇ 2 + V ( r ) ] Ψ ( r ) = E Ψ ( r ) displaystyle left[ frac -hbar ^ 2 2mu nabla ^ 2 +V(mathbf r )right]Psi (mathbf r )=EPsi (mathbf r ) with definitions as above.
The time-independent
Ψ ( t ) displaystyle Psi (t) , then by the linearity of quantum mechanics the wave-function at time t' must be given by Ψ ( t ′ ) = U ( t ′ , t ) Ψ ( t ) displaystyle Psi (t')=U(t',t)Psi (t) , where U ( t ′ , t ) displaystyle U(t',t) is a linear operator. Since time-evolution must preserve the norm of the wave-function, it follows that U ( t ′ , t ) displaystyle U(t',t) must be a member of the unitary group of operators acting on wave-functions. We also know that when t ′ = t displaystyle t'=t , we must have U ( t , t ) = 1 displaystyle U(t,t)=1 . Therefore, expanding the operator U ( t ′ , t ) displaystyle U(t',t) for t' close to t, we can write U ( t ′ , t ) = 1 − i H ( t ′ − t ) displaystyle U(t',t)=1-iH(t'-t) where H is a Hermitian operator. This follows from the fact that the
t ′ − t displaystyle t'-t becomes very small, we obtain Schrödinger's equation.
So far, H is only an abstract Hermitian operator. However using the
correspondence principle it is possible to show that, in the classical
limit, the expectation value of H is indeed the classical energy. The
correspondence principle does not completely fix the form of the
quantum Hamiltonian due to the uncertainty principle and therefore the
precise form of the quantum Hamiltonian must be fixed empirically.
Implications[edit]
The
In classical physics, when a ball is rolled slowly up a large hill, it
will come to a stop and roll back, because it doesn't have enough
energy to get over the top of the hill to the other side. However, the
A double slit experiment showing the accumulation of electrons on a screen as time passes. The nonrelativistic
"If a physical wave field were associated with a particle, or if a particle were identified with a wave packet, then corresponding to N interacting particles there should be N interacting waves in ordinary three-dimensional space. But according to (4.6) that is not the case; instead there is one "wave" function in an abstract 3N-dimensional configuration space. The misinterpretation of psi as a physical wave in ordinary space is possible only because the most common applications of quantum mechanics are to one-particle states, for which configuration space and ordinary space are isomorphic." Two-slit diffraction is a famous example of the strange behaviors that
waves regularly display, that are not intuitively associated with
particles. The overlapping waves from the two slits cancel each other
out in some locations, and reinforce each other in other locations,
causing a complex pattern to emerge. Intuitively, one would not expect
this pattern from firing a single particle at the slits, because the
particle should pass through one slit or the other, not a complex
overlap of both.
However, since the
Erwin Schrödinger Main article: Theoretical and experimental justification for the
Schrödinger equation
Following Max Planck's quantization of light (see black body
radiation),
p = h λ = ℏ k displaystyle p= frac h lambda =hbar k where h is
L = n h 2 π = n ℏ . displaystyle L=n h over 2pi =nhbar . According to de Broglie the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit: n λ = 2 π r . displaystyle nlambda =2pi r., This approach essentially confined the electron wave in one dimension,
along a circular orbit of radius r.
In 1921, prior to de Broglie, Arthur C. Lunn at the University of
Chicago had used the same argument based on the completion of the
relativistic energy–momentum 4-vector to derive what we now call the
de Broglie relation.[12] Unlike de Broglie, Lunn went on to formulate
the differential equation now known as the Schrödinger equation, and
solve for its energy eigenvalues for the hydrogen atom. Unfortunately
the paper was rejected by the Physical Review, as recounted by
Kamen.[13]
Following up on de Broglie's ideas, physicist
i ℏ ∂ ∂ t Ψ ( r , t ) = − ℏ 2 2 m ∇ 2 Ψ ( r , t ) + V ( r ) Ψ ( r , t ) . displaystyle ihbar frac partial partial t Psi (mathbf r ,,t)=- frac hbar ^ 2 2m nabla ^ 2 Psi (mathbf r ,,t)+V(mathbf r )Psi (mathbf r ,,t). However, by that time,
( E + e 2 r ) 2 ψ ( x ) = − ∇ 2 ψ ( x ) + m 2 ψ ( x ) . displaystyle left(E+ e^ 2 over r right)^ 2 psi (x)=-nabla ^ 2 psi (x)+m^ 2 psi (x). He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself in an isolated mountain cabin in December 1925.[18] While at the cabin, Schrödinger decided that his earlier nonrelativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician Hermann Weyl[19]:3) Schrödinger showed that his nonrelativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926.[19]:1[20] In the equation, Schrödinger computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave Ψ(x, t), moving in a potential well V, created by the proton. This computation accurately reproduced the energy levels of the Bohr model. In a paper, Schrödinger himself explained this equation as follows: “ The already ... mentioned psi-function.... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog. ” — Erwin Schrödinger[21] This 1926 paper was enthusiastically endorsed by Einstein, who saw the
matter-waves as an intuitive depiction of nature, as opposed to
Heisenberg's matrix mechanics, which he considered overly formal.[22]
The
“ Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger. ” — Richard Feynman[27] The foundation of the equation is structured to be a linear
differential equation based on classical energy conservation, and
consistent with the De Broglie relations. The solution is the wave
function ψ, which contains all the information that can be known
about the system. In the Copenhagen interpretation, the modulus of ψ
is related to the probability the particles are in some spatial
configuration at some instant of time. Solving the equation for ψ can
be used to predict how the particles will behave under the influence
of the specified potential and with each other.
The
E = T + V = H displaystyle E=T+V=H,! Explicitly, for a particle in one dimension with position x, mass m and momentum p, and potential energy V which generally varies with position and time t: E = p 2 2 m + V ( x , t ) = H . displaystyle E= frac p^ 2 2m +V(x,t)=H. For three dimensions, the position vector r and momentum vector p must be used: E = p ⋅ p 2 m + V ( r , t ) = H displaystyle E= frac mathbf p cdot mathbf p 2m +V(mathbf r ,t)=H This formalism can be extended to any fixed number of particles: the total energy of the system is then the total kinetic energies of the particles, plus the total potential energy, again the Hamiltonian. However, there can be interactions between the particles (an N-body problem), so the potential energy V can change as the spatial configuration of particles changes, and possibly with time. The potential energy, in general, is not the sum of the separate potential energies for each particle, it is a function of all the spatial positions of the particles. Explicitly: E = ∑ n = 1 N p n ⋅ p n 2 m n + V ( r 1 , r 2 ⋯ r N , t ) = H displaystyle E=sum _ n=1 ^ N frac mathbf p _ n cdot mathbf p _ n 2m_ n +V(mathbf r _ 1 ,mathbf r _ 2 cdots mathbf r _ N ,t)=H,! Linearity[edit] The simplest wavefunction is a plane wave of the form: Ψ ( r , t ) = A e i ( k ⋅ r − ω t ) displaystyle Psi (mathbf r ,t)=Ae^ i(mathbf k cdot mathbf r -omega t) ,! where the A is the amplitude, k the wavevector, and ω the angular frequency, of the plane wave. In general, physical situations are not purely described by plane waves, so for generality the superposition principle is required; any wave can be made by superposition of sinusoidal plane waves. So if the equation is linear, a linear combination of plane waves is also an allowed solution. Hence a necessary and separate requirement is that the Schrödinger equation is a linear differential equation. For discrete k the sum is a superposition of plane waves: Ψ ( r , t ) = ∑ n = 1 ∞ A n e i ( k n ⋅ r − ω n t ) displaystyle Psi (mathbf r ,t)=sum _ n=1 ^ infty A_ n e^ i(mathbf k _ n cdot mathbf r -omega _ n t) ,! for some real amplitude coefficients An, and for continuous k the sum
becomes an integral, the
Ψ ( r , t ) = 1 ( 2 π ) 3 ∫ Φ ( k ) e i ( k ⋅ r − ω t ) d 3 k displaystyle Psi (mathbf r ,t)= frac 1 ( sqrt 2pi )^ 3 int Phi (mathbf k )e^ i(mathbf k cdot mathbf r -omega t) d^ 3 mathbf k ,! where d3k = dkxdkydkz is the differential volume element in k-space, and the integrals are taken over all k-space. The momentum wavefunction Φ(k) arises in the integrand since the position and momentum space wavefunctions are Fourier transforms of each other. Consistency with the De Broglie relations[edit] Diagrammatic summary of the quantities related to the wavefunction, as used in De broglie's hypothesis and development of the Schrödinger equation.[28] Einstein's light quanta hypothesis (1905) states that the energy E of a photon is proportional to the frequency ν (or angular frequency, ω = 2πν) of the corresponding quantum wavepacket of light: E = h ν = ℏ ω displaystyle E=hnu =hbar omega ,! Likewise De Broglie's hypothesis (1924) states that any particle can be associated with a wave, and that the momentum p of the particle is inversely proportional to the wavelength λ of such a wave (or proportional to the wavenumber, k = 2π/λ), in one dimension, by: p = h λ = ℏ k , displaystyle p= frac h lambda =hbar k;, while in three dimensions, wavelength λ is related to the magnitude of the wavevector k: p = ℏ k ,
k
= 2 π λ . displaystyle mathbf p =hbar mathbf k ,,quad mathbf k = frac 2pi lambda ,. The Planck–Einstein and de Broglie relations illuminate the deep connections between energy with time, and space with momentum, and express wave–particle duality. In practice, natural units comprising ħ = 1 are used, as the De Broglie equations reduce to identities: allowing momentum, wavenumber, energy and frequency to be used interchangeably, to prevent duplication of quantities, and reduce the number of dimensions of related quantities. For familiarity SI units are still used in this article. Schrödinger's insight,[citation needed] late in 1925, was to express the phase of a plane wave as a complex phase factor using these relations: Ψ = A e i ( k ⋅ r − ω t ) = A e i ( p ⋅ r − E t ) / ℏ displaystyle Psi =Ae^ i(mathbf k cdot mathbf r -omega t) =Ae^ i(mathbf p cdot mathbf r -Et)/hbar and to realize that the first order partial derivatives were: with respect to space: ∇ Ψ = i ℏ p A e i ( p ⋅ r − E t ) / ℏ = i ℏ p Ψ displaystyle nabla Psi = dfrac i hbar mathbf p Ae^ i(mathbf p cdot mathbf r -Et)/hbar = dfrac i hbar mathbf p Psi with respect to time: ∂ Ψ ∂ t = − i E ℏ A e i ( p ⋅ r − E t ) / ℏ = − i E ℏ Ψ displaystyle dfrac partial Psi partial t =- dfrac iE hbar Ae^ i(mathbf p cdot mathbf r -Et)/hbar =- dfrac iE hbar Psi
Another postulate of quantum mechanics is that all observables are
represented by linear
E ^ Ψ = i ℏ ∂ ∂ t Ψ = E Ψ displaystyle hat E Psi =ihbar dfrac partial partial t Psi =EPsi where E are the energy eigenvalues, and the momentum operator, corresponding to the spatial derivatives (the gradient ∇), p ^ Ψ = − i ℏ ∇ Ψ = p Ψ displaystyle hat mathbf p Psi =-ihbar nabla Psi =mathbf p Psi where p is a vector of the momentum eigenvalues. In the above, the "hats" ( ˆ ) indicate these observables are operators, not simply ordinary numbers or vectors. The energy and momentum operators are differential operators, while the potential energy function V is just a multiplicative factor. Substituting the energy and momentum operators into the classical energy conservation equation obtains the operator: E = p ⋅ p 2 m + V → E ^ = p ^ ⋅ p ^ 2 m + V displaystyle E= dfrac mathbf p cdot mathbf p 2m +Vquad rightarrow quad hat E = dfrac hat mathbf p cdot hat mathbf p 2m +V so in terms of derivatives with respect to time and space, acting this operator on the wavefunction Ψ immediately led Schrödinger to his equation:[citation needed] i ℏ ∂ Ψ ∂ t = − ℏ 2 2 m ∇ 2 Ψ + V Ψ displaystyle ihbar dfrac partial Psi partial t =- dfrac hbar ^ 2 2m nabla ^ 2 Psi +VPsi
p ⋅ p ∝ k ⋅ k ∝ T ∝ 1 λ 2 displaystyle mathbf p cdot mathbf p propto mathbf k cdot mathbf k propto Tpropto dfrac 1 lambda ^ 2 The kinetic energy is also proportional to the second spatial derivatives, so it is also proportional to the magnitude of the curvature of the wave, in terms of operators: T ^ Ψ = − ℏ 2 2 m ∇ ⋅ ∇ Ψ ∝ ∇ 2 Ψ . displaystyle hat T Psi = frac -hbar ^ 2 2m nabla cdot nabla Psi ,propto ,nabla ^ 2 Psi ,. As the curvature increases, the amplitude of the wave alternates between positive and negative more rapidly, and also shortens the wavelength. So the inverse relation between momentum and wavelength is consistent with the energy the particle has, and so the energy of the particle has a connection to a wave, all in the same mathematical formulation.[28] Wave and particle motion[edit] Increasing levels of wavepacket localization, meaning the particle has a more localized position. In the limit ħ → 0, the particle's position and momentum become known exactly. This is equivalent to the classical particle. Schrödinger required that a wave packet solution near position r with
wavevector near k will move along the trajectory determined by
classical mechanics for times short enough for the spread in k (and
hence in velocity) not to substantially increase the spread in r.
Since, for a given spread in k, the spread in velocity is proportional
to
σ ( x ) σ ( p x ) ⩾ ℏ 2 → σ ( x ) σ ( p x ) ⩾ 0 displaystyle sigma (x)sigma (p_ x )geqslant frac hbar 2 quad rightarrow quad sigma (x)sigma (p_ x )geqslant 0,! where σ denotes the (root mean square) measurement uncertainty in x and px (and similarly for the y and z directions) which implies the position and momentum can only be known to arbitrary precision in this limit. One simple way to compare classical to quantum mechanics is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the Ehrenfest theorem. For a one-dimensional quantum particle moving in a potential V displaystyle V , the
m d d t ⟨ x ⟩ = ⟨ p ⟩ ; d d t ⟨ p ⟩ = − ⟨ V ′ ( X ) ⟩ . displaystyle m frac d dt langle xrangle =langle prangle ;quad frac d dt langle prangle =-leftlangle V'(X)rightrangle . Although the first of these equations is consistent with the classical behavior, the second is not: If the pair ( ⟨ X ⟩ , ⟨ P ⟩ ) displaystyle (langle Xrangle ,langle Prangle ) were to satisfy Newton's second law, the right-hand side of the second equation would have to be − V ′ ( ⟨ X ⟩ ) displaystyle -V'left(leftlangle Xrightrangle right) , which is typically not the same as − ⟨ V ′ ( X ) ⟩ displaystyle -leftlangle V'(X)rightrangle . In the case of the quantum harmonic oscillator, however, V ′ displaystyle V' is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories. For general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories. If the wave function is highly concentrated around a point x 0 displaystyle x_ 0 , then V ′ ( ⟨ X ⟩ ) displaystyle V'left(leftlangle Xrightrangle right) and ⟨ V ′ ( X ) ⟩ displaystyle leftlangle V'(X)rightrangle will be almost the same, since both will be approximately equal to V ′ ( x 0 ) displaystyle V'(x_ 0 ) . In that case, the expected position and expected momentum will
remain very close to the classical trajectories, at least for as long
as the wave function remains highly localized in position.[34] When
i ℏ ∂ ∂ t Ψ ( r , t ) = H ^ Ψ ( r , t ) displaystyle ihbar frac partial partial t Psi left(mathbf r ,tright)= hat H Psi left(mathbf r ,tright),! is closely related to the
− ∂ ∂ t S ( q i , t ) = H ( q i , ∂ S ∂ q i , t ) displaystyle - frac partial partial t S(q_ i ,t)=Hleft(q_ i , frac partial S partial q_ i ,tright),! where S is action and H is the Hamiltonian function (not operator). Here the generalized coordinates qi for i = 1, 2, 3 (used in the context of the HJE) can be set to the position in Cartesian coordinates as r = (q1, q2, q3) = (x, y, z).[32] Substituting Ψ = ρ ( r , t ) e i S ( r , t ) / ℏ displaystyle Psi = sqrt rho (mathbf r ,t) e^ iS(mathbf r ,t)/hbar ,! where ρ is the probability density, into the Schrödinger equation and then taking the limit ħ → 0 in the resulting equation, yields the Hamilton–Jacobi equation. The implications are as follows: The motion of a particle, described by a (short-wavelength) wave
packet solution to the Schrödinger equation, is also described by the
Nonrelativistic quantum mechanics[edit]
The quantum mechanics of particles without accounting for the effects
of special relativity, for example particles propagating at speeds
much less than light, is known as nonrelativistic quantum mechanics.
Following are several forms of Schrödinger's equation in this context
for different situations: time independence and dependence, one and
three spatial dimensions, and one and N particles.
In actuality, the particles constituting the system do not have the
numerical labels used in theory. The language of mathematics forces us
to label the positions of particles one way or another, otherwise
there would be confusion between symbols representing which variables
are for which particle.[30]
Ψ ( space coords , t ) = ψ ( space coords ) τ ( t ) . displaystyle Psi ( text space coords ,t)=psi ( text space coords )tau (t),. where ψ(space coords) is a function of all the spatial coordinate(s)
of the particle(s) constituting the system only, and τ(t) is a
function of time only.
Substituting for ψ into the
Ψ ( space coords , t ) = ψ ( space coords ) e − i E t / ℏ . displaystyle Psi ( text space coords ,t)=psi ( text space coords )e^ -i Et/hbar ,. Since the time dependent phase factor is always the same, only the
spatial part needs to be solved for in time independent problems.
Additionally, the energy operator Ê = iħ∂/∂t can always be
replaced by the energy eigenvalue E, thus the time independent
H ^ ψ = E ψ displaystyle hat H psi =Epsi This is true for any number of particles in any number of dimensions (in a time independent potential). This case describes the standing wave solutions of the time-dependent equation, which are the states with definite energy (instead of a probability distribution of different energies). In physics, these standing waves are called "stationary states" or "energy eigenstates"; in chemistry they are called "atomic orbitals" or "molecular orbitals". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels. The energy eigenvalues from this equation form a discrete spectrum of values, so mathematically energy must be quantized. More specifically, the energy eigenstates form a basis – any wavefunction may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is the spectral theorem in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix. One-dimensional examples[edit] For a particle in one dimension, the Hamiltonian is: H ^ = p ^ 2 2 m + V ( x ) , p ^ = − i ℏ d d x displaystyle hat H = frac hat p ^ 2 2m +V(x),,quad hat p =-ihbar frac d dx and substituting this into the general
− ℏ 2 2 m d 2 d x 2 ψ ( x ) + V ( x ) ψ ( x ) = E ψ ( x ) displaystyle - frac hbar ^ 2 2m frac d^ 2 dx^ 2 psi (x)+V(x)psi (x)=Epsi (x) This is the only case the
Ψ ( x , t ) = ψ ( x ) e − i E t / ℏ . displaystyle Psi (x,t)=psi (x)e^ -iEt/hbar ,. For N particles in one dimension, the Hamiltonian is: H ^ = ∑ n = 1 N p ^ n 2 2 m n + V ( x 1 , x 2 , ⋯ x N ) , p ^ n = − i ℏ ∂ ∂ x n displaystyle hat H =sum _ n=1 ^ N frac hat p _ n ^ 2 2m_ n +V(x_ 1 ,x_ 2 ,cdots x_ N ),,quad hat p _ n =-ihbar frac partial partial x_ n where the position of particle n is xn. The corresponding Schrödinger equation is: − ℏ 2 2 ∑ n = 1 N 1 m n ∂ 2 ∂ x n 2 ψ ( x 1 , x 2 , ⋯ x N ) + V ( x 1 , x 2 , ⋯ x N ) ψ ( x 1 , x 2 , ⋯ x N ) = E ψ ( x 1 , x 2 , ⋯ x N ) . displaystyle - frac hbar ^ 2 2 sum _ n=1 ^ N frac 1 m_ n frac partial ^ 2 partial x_ n ^ 2 psi (x_ 1 ,x_ 2 ,cdots x_ N )+V(x_ 1 ,x_ 2 ,cdots x_ N )psi (x_ 1 ,x_ 2 ,cdots x_ N )=Epsi (x_ 1 ,x_ 2 ,cdots x_ N ),. so the general solutions have the form: Ψ ( x 1 , x 2 , ⋯ x N , t ) = e − i E t / ℏ ψ ( x 1 , x 2 ⋯ x N ) displaystyle Psi (x_ 1 ,x_ 2 ,cdots x_ N ,t)=e^ -iEt/hbar psi (x_ 1 ,x_ 2 cdots x_ N ) For non-interacting distinguishable particles,[35] the potential of the system only influences each particle separately, so the total potential energy is the sum of potential energies for each particle: V ( x 1 , x 2 , ⋯ x N ) = ∑ n = 1 N V ( x n ) . displaystyle V(x_ 1 ,x_ 2 ,cdots x_ N )=sum _ n=1 ^ N V(x_ n ),. and the wavefunction can be written as a product of the wavefunctions for each particle: Ψ ( x 1 , x 2 , ⋯ x N , t ) = e − i E t / ℏ ∏ n = 1 N ψ ( x n ) , displaystyle Psi (x_ 1 ,x_ 2 ,cdots x_ N ,t)=e^ -i Et/hbar prod _ n=1 ^ N psi (x_ n ),, For non-interacting identical particles, the potential is still a sum, but wavefunction is a bit more complicated – it is a sum over the permutations of products of the separate wavefunctions to account for particle exchange. In general for interacting particles, the above decompositions are not possible. Free particle[edit] For no potential, V = 0, so the particle is free and the equation reads:[5]:151ff − E ψ = ℏ 2 2 m d 2 ψ d x 2 displaystyle -Epsi = frac hbar ^ 2 2m d^ 2 psi over dx^ 2 , which has oscillatory solutions for E > 0 (the Cn are arbitrary constants): ψ E ( x ) = C 1 e i 2 m E / ℏ 2 x + C 2 e − i 2 m E / ℏ 2 x displaystyle psi _ E (x)=C_ 1 e^ i sqrt 2mE/hbar ^ 2 ,x +C_ 2 e^ -i sqrt 2mE/hbar ^ 2 ,x , and exponential solutions for E < 0 ψ −
E
( x ) = C 1 e 2 m
E
/ ℏ 2 x + C 2 e − 2 m
E
/ ℏ 2 x . displaystyle psi _ -E (x)=C_ 1 e^ sqrt 2mE/hbar ^ 2 ,x +C_ 2 e^ - sqrt 2mE/hbar ^ 2 ,x ., The exponentially growing solutions have an infinite norm, and are not physical. They are not allowed in a finite volume with periodic or fixed boundary conditions. See also free particle and wavepacket for more discussion on the free particle. Constant potential[edit] For a constant potential, V = V0, the solution is oscillatory for E > V0 and exponential for E < V0, corresponding to energies that are allowed or disallowed in classical mechanics. Oscillatory solutions have a classically allowed energy and correspond to actual classical motions, while the exponential solutions have a disallowed energy and describe a small amount of quantum bleeding into the classically disallowed region, due to quantum tunneling. If the potential V0 grows to infinity, the motion is classically confined to a finite region. Viewed far enough away, every solution is reduced to an exponential; the condition that the exponential is decreasing restricts the energy levels to a discrete set, called the allowed energies.[31] Harmonic oscillator[edit] A harmonic oscillator in classical mechanics (A–B) and quantum
mechanics (C–H). In (A–B), a ball, attached to a spring,
oscillates back and forth. (C–H) are six solutions to the
Schrödinger
Main article:
E ψ = − ℏ 2 2 m d 2 d x 2 ψ + 1 2 m ω 2 x 2 ψ displaystyle Epsi =- frac hbar ^ 2 2m frac d^ 2 dx^ 2 psi + frac 1 2 momega ^ 2 x^ 2 psi It is a notable quantum system to solve for; since the solutions are exact (but complicated – in terms of Hermite polynomials), and it can describe or at least approximate a wide variety of other systems, including vibrating atoms, molecules,[36] and atoms or ions in lattices,[37] and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics. There is a family of solutions – in the position basis they are ψ n ( x ) = 1 2 n n ! ⋅ ( m ω π ℏ ) 1 / 4 ⋅ e − m ω x 2 2 ℏ ⋅ H n ( m ω ℏ x ) displaystyle psi _ n (x)= sqrt frac 1 2^ n ,n! cdot left( frac momega pi hbar right)^ 1/4 cdot e^ - frac momega x^ 2 2hbar cdot H_ n left( sqrt frac momega hbar xright) where n = 0,1,2,..., and the functions Hn are the Hermite polynomials. Three-dimensional examples[edit] The extension from one dimension to three dimensions is straightforward, all position and momentum operators are replaced by their three-dimensional expressions and the partial derivative with respect to space is replaced by the gradient operator. The Hamiltonian for one particle in three dimensions is: H ^ = p ^ ⋅ p ^ 2 m + V ( r ) , p ^ = − i ℏ ∇ displaystyle hat H = frac hat mathbf p cdot hat mathbf p 2m +V(mathbf r ),,quad hat mathbf p =-ihbar nabla generating the equation: − ℏ 2 2 m ∇ 2 ψ ( r ) + V ( r ) ψ ( r ) = E ψ ( r ) displaystyle - frac hbar ^ 2 2m nabla ^ 2 psi (mathbf r )+V(mathbf r )psi (mathbf r )=Epsi (mathbf r ) with stationary state solutions of the form: Ψ ( r , t ) = ψ ( r ) e − i E t / ℏ displaystyle Psi (mathbf r ,t)=psi (mathbf r )e^ -iEt/hbar where the position of the particle is r. Two useful coordinate systems
for solving the
H ^ = ∑ n = 1 N p ^ n ⋅ p ^ n 2 m n + V ( r 1 , r 2 , ⋯ r N ) , p ^ n = − i ℏ ∇ n displaystyle hat H =sum _ n=1 ^ N frac hat mathbf p _ n cdot hat mathbf p _ n 2m_ n +V(mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N ),,quad hat mathbf p _ n =-ihbar nabla _ n where the position of particle n is rn and the gradient operators are
partial derivatives with respect to the particle's position
coordinates. In Cartesian coordinates, for particle n, the position
vector is rn = (xn, yn, zn) while the gradient and
∇ n = e x ∂ ∂ x n + e y ∂ ∂ y n + e z ∂ ∂ z n , ∇ n 2 = ∇ n ⋅ ∇ n = ∂ 2 ∂ x n 2 + ∂ 2 ∂ y n 2 + ∂ 2 ∂ z n 2 displaystyle nabla _ n =mathbf e _ x frac partial partial x_ n +mathbf e _ y frac partial partial y_ n +mathbf e _ z frac partial partial z_ n ,,quad nabla _ n ^ 2 =nabla _ n cdot nabla _ n = frac partial ^ 2 partial x_ n ^ 2 + frac partial ^ 2 partial y_ n ^ 2 + frac partial ^ 2 partial z_ n ^ 2 The
− ℏ 2 2 ∑ n = 1 N 1 m n ∇ n 2 Ψ ( r 1 , r 2 , ⋯ r N ) + V ( r 1 , r 2 , ⋯ r N ) Ψ ( r 1 , r 2 , ⋯ r N ) = E Ψ ( r 1 , r 2 , ⋯ r N ) displaystyle - frac hbar ^ 2 2 sum _ n=1 ^ N frac 1 m_ n nabla _ n ^ 2 Psi (mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N )+V(mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N )Psi (mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N )=EPsi (mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N ) with stationary state solutions: Ψ ( r 1 , r 2 ⋯ r N , t ) = e − i E t / ℏ ψ ( r 1 , r 2 ⋯ r N ) displaystyle Psi (mathbf r _ 1 ,mathbf r _ 2 cdots mathbf r _ N ,t)=e^ -iEt/hbar psi (mathbf r _ 1 ,mathbf r _ 2 cdots mathbf r _ N ) Again, for non-interacting distinguishable particles the potential is the sum of particle potentials V ( r 1 , r 2 , ⋯ r N ) = ∑ n = 1 N V ( r n ) displaystyle V(mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N )=sum _ n=1 ^ N V(mathbf r _ n ) and the wavefunction is a product of the particle wavefunctions Ψ ( r 1 , r 2 ⋯ r N , t ) = e − i E t / ℏ ∏ n = 1 N ψ ( r n ) . displaystyle Psi (mathbf r _ 1 ,mathbf r _ 2 cdots mathbf r _ N ,t)=e^ -i Et/hbar prod _ n=1 ^ N psi (mathbf r _ n ),. For non-interacting identical particles, the potential is a sum but
the wavefunction is a sum over permutations of products. The previous
two equations do not apply to interacting particles.
Following are examples where exact solutions are known. See the main
articles for further details.
E ψ = − ℏ 2 2 μ ∇ 2 ψ − e 2 4 π ε 0 r ψ displaystyle Epsi =- frac hbar ^ 2 2mu nabla ^ 2 psi - frac e^ 2 4pi varepsilon _ 0 r psi where e is the electron charge, r is the position of the electron (r = r is the magnitude of the position), the potential term is due to the Coulomb interaction, wherein ε0 is the electric constant (permittivity of free space) and μ = m e m p m e + m p displaystyle mu = frac m_ e m_ p m_ e +m_ p is the 2-body reduced mass of the hydrogen nucleus (just a proton) of mass mp and the electron of mass me. The negative sign arises in the potential term since the proton and electron are oppositely charged. The reduced mass in place of the electron mass is used since the electron and proton together orbit each other about a common centre of mass, and constitute a two-body problem to solve. The motion of the electron is of principle interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass. The wavefunction for hydrogen is a function of the electron's coordinates, and in fact can be separated into functions of each coordinate.[38] Usually this is done in spherical polar coordinates: ψ ( r , θ , ϕ ) = R ( r ) Y ℓ m ( θ , ϕ ) = R ( r ) Θ ( θ ) Φ ( ϕ ) displaystyle psi (r,theta ,phi )=R(r)Y_ ell ^ m (theta ,phi )=R(r)Theta (theta )Phi (phi ) where R are radial functions and Ym
ℓ(θ, φ) are spherical harmonics of degree ℓ and order m. This is
the only atom for which the
ψ n ℓ m ( r , θ , ϕ ) = ( 2 n a 0 ) 3 ( n − ℓ − 1 ) ! 2 n [ ( n + ℓ ) ! ] e − r / n a 0 ( 2 r n a 0 ) ℓ L n − ℓ − 1 2 ℓ + 1 ( 2 r n a 0 ) ⋅ Y ℓ m ( θ , ϕ ) displaystyle psi _ nell m (r,theta ,phi )= sqrt left( frac 2 na_ 0 right) ^ 3 frac (n-ell -1)! 2n[(n+ell )!] e^ -r/na_ 0 left( frac 2r na_ 0 right)^ ell L_ n-ell -1 ^ 2ell +1 left( frac 2r na_ 0 right)cdot Y_ ell ^ m (theta ,phi ) where: a 0 = 4 π ε 0 ℏ 2 m e e 2 displaystyle a_ 0 = frac 4pi varepsilon _ 0 hbar ^ 2 m_ e e^ 2 is the Bohr radius, L n − ℓ − 1 2 ℓ + 1 ( ⋯ ) displaystyle L_ n-ell -1 ^ 2ell +1 (cdots ) are the generalized Laguerre polynomials of degree n − ℓ − 1. n, ℓ, m are the principal, azimuthal, and magnetic quantum numbers respectively: which take the values: n = 1 , 2 , 3 , … ℓ = 0 , 1 , 2 , … , n − 1 m = − ℓ , … , ℓ displaystyle begin aligned n&=1,2,3,dots \ell &=0,1,2,dots ,n-1\m&=-ell ,dots ,ell \end aligned NB: generalized Laguerre polynomials are defined differently by different authors—see main article on them and the hydrogen atom. Two-electron atoms or ions[edit] The equation for any two-electron system, such as the neutral helium atom (He, Z = 2), the negative hydrogen ion (H−, Z = 1), or the positive lithium ion (Li+, Z = 3) is:[29] E ψ = − ℏ 2 [ 1 2 μ ( ∇ 1 2 + ∇ 2 2 ) + 1 M ∇ 1 ⋅ ∇ 2 ] ψ + e 2 4 π ε 0 [ 1 r 12 − Z ( 1 r 1 + 1 r 2 ) ] ψ displaystyle Epsi =-hbar ^ 2 left[ frac 1 2mu left(nabla _ 1 ^ 2 +nabla _ 2 ^ 2 right)+ frac 1 M nabla _ 1 cdot nabla _ 2 right]psi + frac e^ 2 4pi varepsilon _ 0 left[ frac 1 r_ 12 -Zleft( frac 1 r_ 1 + frac 1 r_ 2 right)right]psi where r1 is the position of one electron (r1 = r1 is its magnitude), r2 is the position of the other electron (r2 = r2 is the magnitude), r12 = r12 is the magnitude of the separation between them given by
r 12
=
r 2 − r 1
displaystyle mathbf r _ 12 =mathbf r _ 2 -mathbf r _ 1 ,! μ is again the two-body reduced mass of an electron with respect to the nucleus of mass M, so this time μ = m e M m e + M displaystyle mu = frac m_ e M m_ e +M ,! and Z is the atomic number for the element (not a quantum number). The cross-term of two laplacians 1 M ∇ 1 ⋅ ∇ 2 displaystyle frac 1 M nabla _ 1 cdot nabla _ 2 ,! is known as the mass polarization term, which arises due to the motion of atomic nuclei. The wavefunction is a function of the two electron's positions: ψ = ψ ( r 1 , r 2 ) . displaystyle psi =psi (mathbf r _ 1 ,mathbf r _ 2 ). There is no closed form solution for this equation.
i ℏ ∂ ∂ t Ψ = H ^ Ψ . displaystyle ihbar frac partial partial t Psi = hat H Psi . and the solution, the wavefunction, is a function of all the particle coordinates of the system and time. Following are specific cases. For one particle in one dimension, the Hamiltonian H ^ = p ^ 2 2 m + V ( x , t ) , p ^ = − i ℏ ∂ ∂ x displaystyle hat H = frac hat p ^ 2 2m +V(x,t),,quad hat p =-ihbar frac partial partial x generates the equation: i ℏ ∂ ∂ t Ψ ( x , t ) = − ℏ 2 2 m ∂ 2 ∂ x 2 Ψ ( x , t ) + V ( x , t ) Ψ ( x , t ) displaystyle ihbar frac partial partial t Psi (x,t)=- frac hbar ^ 2 2m frac partial ^ 2 partial x^ 2 Psi (x,t)+V(x,t)Psi (x,t) For N particles in one dimension, the Hamiltonian is: H ^ = ∑ n = 1 N p ^ n 2 2 m n + V ( x 1 , x 2 , ⋯ x N , t ) , p ^ n = − i ℏ ∂ ∂ x n displaystyle hat H =sum _ n=1 ^ N frac hat p _ n ^ 2 2m_ n +V(x_ 1 ,x_ 2 ,cdots x_ N ,t),,quad hat p _ n =-ihbar frac partial partial x_ n where the position of particle n is xn, generating the equation: i ℏ ∂ ∂ t Ψ ( x 1 , x 2 ⋯ x N , t ) = − ℏ 2 2 ∑ n = 1 N 1 m n ∂ 2 ∂ x n 2 Ψ ( x 1 , x 2 ⋯ x N , t ) + V ( x 1 , x 2 ⋯ x N , t ) Ψ ( x 1 , x 2 ⋯ x N , t ) . displaystyle ihbar frac partial partial t Psi (x_ 1 ,x_ 2 cdots x_ N ,t)=- frac hbar ^ 2 2 sum _ n=1 ^ N frac 1 m_ n frac partial ^ 2 partial x_ n ^ 2 Psi (x_ 1 ,x_ 2 cdots x_ N ,t)+V(x_ 1 ,x_ 2 cdots x_ N ,t)Psi (x_ 1 ,x_ 2 cdots x_ N ,t),. For one particle in three dimensions, the Hamiltonian is: H ^ = p ^ ⋅ p ^ 2 m + V ( r , t ) , p ^ = − i ℏ ∇ displaystyle hat H = frac hat mathbf p cdot hat mathbf p 2m +V(mathbf r ,t),,quad hat mathbf p =-ihbar nabla generating the equation: i ℏ ∂ ∂ t Ψ ( r , t ) = − ℏ 2 2 m ∇ 2 Ψ ( r , t ) + V ( r , t ) Ψ ( r , t ) displaystyle ihbar frac partial partial t Psi (mathbf r ,t)=- frac hbar ^ 2 2m nabla ^ 2 Psi (mathbf r ,t)+V(mathbf r ,t)Psi (mathbf r ,t) For N particles in three dimensions, the Hamiltonian is: H ^ = ∑ n = 1 N p ^ n ⋅ p ^ n 2 m n + V ( r 1 , r 2 , ⋯ r N , t ) , p ^ n = − i ℏ ∇ n displaystyle hat H =sum _ n=1 ^ N frac hat mathbf p _ n cdot hat mathbf p _ n 2m_ n +V(mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N ,t),,quad hat mathbf p _ n =-ihbar nabla _ n where the position of particle n is rn, generating the equation:[5]:141 i ℏ ∂ ∂ t Ψ ( r 1 , r 2 , ⋯ r N , t ) = − ℏ 2 2 ∑ n = 1 N 1 m n ∇ n 2 Ψ ( r 1 , r 2 , ⋯ r N , t ) + V ( r 1 , r 2 , ⋯ r N , t ) Ψ ( r 1 , r 2 , ⋯ r N , t ) displaystyle ihbar frac partial partial t Psi (mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N ,t)=- frac hbar ^ 2 2 sum _ n=1 ^ N frac 1 m_ n nabla _ n ^ 2 Psi (mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N ,t)+V(mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N ,t)Psi (mathbf r _ 1 ,mathbf r _ 2 ,cdots mathbf r _ N ,t) This last equation is in a very high dimension, so the solutions are not easy to visualize. Solution methods[edit] This article is in a list format that may be better presented using prose. You can help by converting this article to prose, if appropriate. Editing help is available. (October 2016) General techniques: Perturbation theory
The variational method
Methods for special cases: List of quantum-mechanical systems with analytical solutions
Properties[edit]
The
ψ = a ψ 1 + b ψ 2 displaystyle displaystyle psi =apsi _ 1 +bpsi _ 2 where a and b are any complex numbers (the sum can be extended for any
number of wavefunctions). This property allows superpositions of
quantum states to be solutions of the Schrödinger equation. Even more
generally, it holds that a general solution to the Schrödinger
equation can be found by taking a weighted sum over all single state
solutions achievable. For example, consider a wave function Ψ(x, t)
such that the wave function is a product of two functions: one time
independent, and one time dependent. If states of definite energy
found using the time independent
e − i E n t / ℏ , displaystyle e^ -iE_ n t /hbar , then a valid general solution is Ψ ( x , t ) = ∑ n A n ψ E n ( x ) e − i E n t / ℏ . displaystyle displaystyle Psi (x,t)=sum limits _ n A_ n psi _ E_ n (x)e^ -iE_ n t /hbar . Additionally, the ability to scale solutions allows one to solve for a wave function without normalizing it first. If one has a set of normalized solutions ψn, then Ψ = ∑ n A n ψ n displaystyle displaystyle Psi =sum limits _ n A_ n psi _ n can be normalized by ensuring that ∑ n
A n
2 = 1. displaystyle displaystyle sum limits _ n A_ n ^ 2 =1. This is much more convenient than having to verify that ∫ − ∞ ∞
Ψ ( x )
2 d x = ∫ − ∞ ∞ Ψ ( x ) Ψ ∗ ( x ) d x = 1. displaystyle displaystyle int limits _ -infty ^ infty Psi (x)^ 2 ,dx=int limits _ -infty ^ infty Psi (x)Psi ^ * (x),dx=1.
i ℏ ∂ t
ψ ⟩ = H ^
ψ ⟩ displaystyle ihbar partial _ t psi rangle = hat H psi rangle is often presented in the position basis form i ℏ ∂ t ψ ( x ) = − ℏ 2 2 m ∇ 2 ψ ( x ) + V ( x ) ψ ( x ) displaystyle ihbar partial _ t psi (x)=- frac hbar ^ 2 2m nabla ^ 2 psi (x)+V(x)psi (x) (with ⟨ x
ψ ⟩ ≡ ψ ( x ) displaystyle langle xpsi rangle equiv psi (x) ). But as a vector operator equation it has a valid representation in any arbitrary complete basis of kets in Hilbert space. For example, in the momentum space basis the equation reads i ℏ ∂ t f ( p ) = p 2 2 m f ( p ) + ( V ~ ∗ f ) ( p ) displaystyle displaystyle ihbar partial _ t f(p)= frac p^ 2 2m f(p)+( tilde V *f)(p) where
p ⟩ displaystyle prangle is the plane wave state of definite momentum p displaystyle p , ⟨ p
ψ ⟩ ≡ f ( p ) = ∫ ψ ( x ) e − i p x d x displaystyle langle ppsi rangle equiv f(p)=int psi (x)e^ -ipx dx , V ~ displaystyle tilde V is the
V displaystyle V , and ∗ displaystyle * denotes Fourier convolution. In the 1D example with absence of a potential, V ~ = 0 displaystyle tilde V =0 (or similarly V ~ ( k ) ∝ δ ( k ) displaystyle tilde V (k)propto delta (k) in the case of a background potential constant throughout space), each stationary state of energy ℏ ω = q 2 / 2 m displaystyle hbar omega =q^ 2 /2m is of the form f q ( p ) = ( c + δ ( p − q ) + c − δ ( p + q ) ) e − i ω t displaystyle f_ q (p)=(c_ + delta (p-q)+c_ - delta (p+q))e^ -iomega t for arbitrary complex coefficients c ± displaystyle c_ pm . Such a wave function, as expected in free space, is a superposition of plane waves moving right and left with momenta ± q displaystyle pm q ; upon momentum measurement the state would collapse to one of definite momentum ± q displaystyle pm q with probability ∝
c ±
2 displaystyle propto c_ pm ^ 2 .
A version of the momentum space
f ( p ) displaystyle f(p) with f ( p + K ) displaystyle f(p+K) for only discrete reciprocal lattice vectors K displaystyle K . This makes it convenient to solve the momentum space Schrödinger
equation at each point in the
H ^ ( a ψ 1 + b ψ 2 ) = a H ^ ψ 1 + b H ^ ψ 2 = E ( a ψ 1 + b ψ 2 ) . displaystyle hat H (apsi _ 1 +bpsi _ 2 )=a hat H psi _ 1 +b hat H psi _ 2 =E(apsi _ 1 +bpsi _ 2 ). Two different solutions with the same energy are called degenerate.[31] In an arbitrary potential, if a wave function ψ solves the time-independent equation, so does its complex conjugate, denoted ψ*. By taking linear combinations, the real and imaginary parts of ψ are each solutions. If there is no degeneracy they can only differ by a factor. In the time-dependent equation, complex conjugate waves move in opposite directions. If Ψ(x, t) is one solution, then so is Ψ*(x, –t). The symmetry of complex conjugation is called time-reversal symmetry. Space and time derivatives[edit] Continuity of the wavefunction and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t. The
i ℏ ∂ Ψ ∂ t = − ℏ 2 2 m ( ∂ 2 Ψ ∂ x 2 + ∂ 2 Ψ ∂ y 2 + ∂ 2 Ψ ∂ z 2 ) + V ( x , y , z , t ) Ψ . displaystyle ihbar partial Psi over partial t =- hbar ^ 2 over 2m left( partial ^ 2 Psi over partial x^ 2 + partial ^ 2 Psi over partial y^ 2 + partial ^ 2 Psi over partial z^ 2 right)+V(x,y,z,t)Psi .,! The first time partial derivative implies the initial value (at t = 0) of the wavefunction Ψ ( x , y , z , 0 ) displaystyle Psi (x,y,z,0),! is an arbitrary constant. Likewise – the second order derivatives with respect to space implies the wavefunction and its first order spatial derivatives Ψ ( x b , y b , z b , t ) ∂ ∂ x Ψ ( x b , y b , z b , t ) ∂ ∂ y Ψ ( x b , y b , z b , t ) ∂ ∂ z Ψ ( x b , y b , z b , t ) displaystyle begin aligned &Psi (x_ b ,y_ b ,z_ b ,t)\& frac partial partial x Psi (x_ b ,y_ b ,z_ b ,t)quad frac partial partial y Psi (x_ b ,y_ b ,z_ b ,t)quad frac partial partial z Psi (x_ b ,y_ b ,z_ b ,t)end aligned ,! are all arbitrary constants at a given set of points, where xb, yb, zb
are a set of points describing boundary b (derivatives are evaluated
at the boundaries). Typically there are one or two boundaries, such as
the step potential and particle in a box respectively.
As the first order derivatives are arbitrary, the wavefunction can be
a continuously differentiable function of space, since at any boundary
the gradient of the wavefunction can be matched.
On the contrary, wave equations in physics are usually second order in
time, notable are the family of classical wave equations and the
quantum Klein–Gordon equation.
Local conservation of probability[edit]
Main articles:
∂ ∂ t ρ ( r , t ) + ∇ ⋅ j = 0 , displaystyle partial over partial t rho left(mathbf r ,tright)+nabla cdot mathbf j =0, where ρ =
Ψ
2 = Ψ ∗ ( r , t ) Ψ ( r , t ) displaystyle rho =Psi ^ 2 =Psi ^ * (mathbf r ,t)Psi (mathbf r ,t),! is the probability density (probability per unit volume, * denotes complex conjugate), and j = 1 2 m ( Ψ ∗ p ^ Ψ − Ψ p ^ Ψ ∗ ) displaystyle mathbf j = 1 over 2m left(Psi ^ * hat mathbf p Psi -Psi hat mathbf p Psi ^ * right),! is the probability current (flow per unit area).
Hence predictions from the
⟨ ψ
A ^
ψ ⟩ displaystyle langle psi hat A psi rangle over all ψ which are normalized.[40] In this way, the smallest eigenvalue is expressed through the variational principle. For the Schrödinger Hamiltonian Ĥ bounded from below, the smallest eigenvalue is called the ground state energy. That energy is the minimum value of ⟨ ψ
H ^
ψ ⟩ = ∫ ψ ∗ ( r ) [ − ℏ 2 2 m ∇ 2 ψ ( r ) + V ( r ) ψ ( r ) ] d 3 r = ∫ [ ℏ 2 2 m
∇ ψ
2 + V ( r )
ψ
2 ] d 3 r = ⟨ H ^ ⟩ displaystyle langle psi hat H psi rangle =int psi ^ * (mathbf r )left[- frac hbar ^ 2 2m nabla ^ 2 psi (mathbf r )+V(mathbf r )psi (mathbf r )right]d^ 3 mathbf r =int left[ frac hbar ^ 2 2m nabla psi ^ 2 +V(mathbf r )psi ^ 2 right]d^ 3 mathbf r =langle hat H rangle (using integration by parts). Due to the complex modulus of ψ2 (which
is positive definite), the right hand side is always greater than the
lowest value of V(x). In particular, the ground state energy is
positive when V(x) is everywhere positive.
For potentials which are bounded below and are not infinite over a
region, there is a ground state which minimizes the integral above.
This lowest energy wavefunction is real and positive definite –
meaning the wavefunction can increase and decrease, but is positive
for all positions. It physically cannot be negative: if it were,
smoothing out the bends at the sign change (to minimize the
wavefunction) rapidly reduces the gradient contribution to the
integral and hence the kinetic energy, while the potential energy
changes linearly and less quickly. The kinetic and potential energy
are both changing at different rates, so the total energy is not
constant, which can't happen (conservation). The solutions are
consistent with
∂ ∂ τ X ( r , τ ) = ℏ 2 m ∇ 2 X ( r , τ ) , X ( r , τ ) = Ψ ( r , τ / i ) displaystyle partial over partial tau X(mathbf r ,tau )= frac hbar 2m nabla ^ 2 X(mathbf r ,tau ),,quad X(mathbf r ,tau )=Psi (mathbf r ,tau /i) which has the same form as the diffusion equation, with diffusion coefficient ħ/2m. In that case, the diffusivity yields the De Broglie relation in accordance with the Markov process.[42] Regularity[edit] On the space L 2 displaystyle L^ 2 of square-integrable densities, the Schrödinger semigroup e i t H ^ displaystyle e^ it hat H is a unitary evolution, and therefore surjective. The flows satisfy
the
i ∂ t u = H ^ u displaystyle ipartial _ t u= widehat H u , where the derivative is taken in the distribution sense. However, since H ^ displaystyle widehat H for most physically reasonable Hamiltonians (e.g., the Laplace operator, possibly modified by a potential) is unbounded in L 2 displaystyle L^ 2 , this shows that the semigroup flows lack Sobolev regularity in
general. Instead, solutions of the
E 2 = ( p c ) 2 + ( m 0 c 2 ) 2 , displaystyle E^ 2 =(pc)^ 2 +(m_ 0 c^ 2 )^ 2 ,, instead of classical energy equations. The
1 c 2 ∂ 2 ∂ t 2 ψ − ∇ 2 ψ + m 2 c 2 ℏ 2 ψ = 0. displaystyle frac 1 c^ 2 frac partial ^ 2 partial t^ 2 psi -nabla ^ 2 psi + frac m^ 2 c^ 2 hbar ^ 2 psi =0. , was the first such equation to be obtained, even before the
nonrelativistic one, and applies to massive spinless particles. The
( β m c 2 + c ( ∑ n = 1 3 α n p n ) ) ψ = i ℏ ∂ ψ ∂ t displaystyle left(beta mc^ 2 +cleft(sum _ nmathop = 1 ^ 3 alpha _ n p_ n right)right)psi =ihbar frac partial psi partial t The general form of the
H ^ Dirac = γ 0 [ c γ ⋅ ( p ^ − q A ) + m c 2 + γ 0 q ϕ ] , displaystyle hat H _ text Dirac =gamma ^ 0 left[c boldsymbol gamma cdot left( hat mathbf p -qmathbf A right)+mc^ 2 +gamma ^ 0 qphi right],, in which the γ = (γ1, γ2, γ3) and γ0 are the Dirac gamma matrices
related to the spin of the particle. The
− i ∂ z ψ = ( i η ∂ t + η † m ) ψ displaystyle begin aligned -ipartial _ z psi =(ieta partial _ t +eta ^ dagger m)psi end aligned This equation allows for the inclusion of spin in nonrelativistic quantum mechanics. Squaring the above equation yields the Schrödinger equation in 1D. The matrices η displaystyle eta obey the following properties η 2 = 0 ( η † ) 2 = 0 η , η † = 2 I displaystyle begin aligned eta ^ 2 =0\(eta ^ dagger )^ 2 =0\leftlbrace eta ,eta ^ dagger rightrbrace =2Iend aligned The 3 dimensional version of the equation is given by − i γ i ∂ i ψ = ( i η ∂ t + η † m ) ψ displaystyle begin aligned -igamma _ i partial _ i psi =(ieta partial _ t +eta ^ dagger m)psi end aligned Here η = ( γ 0 + i γ 5 ) / 2 displaystyle eta =(gamma _ 0 +igamma _ 5 )/ sqrt 2 is a 4 × 4 displaystyle 4times 4 nilpotent matrix and γ i displaystyle gamma _ i are the Dirac gamma matrices ( i = 1 , 2 , 3 displaystyle i=1,2,3 ). The
E − m ≃ E ′ displaystyle E-msimeq E' and E + m ≃ 2 m displaystyle E+msimeq 2m , the above equation can be derived from the Dirac equation.[44] See also[edit] Eckhaus equation
Fractional Schrödinger equation
List of quantum-mechanical systems with analytical solutions
Logarithmic Schrödinger equation
Nonlinear Schrödinger equation
Notes[edit] ^ "Physicist Erwin Schrödinger's Google doodle marks quantum
mechanics work". The Guardian. 13 August 2013. Retrieved 25 August
2013.
^ Schrödinger, E. (1926). "An Undulatory Theory of the
References[edit] P. A. M. Dirac (1958). The Principles of
External links[edit] Hazewinkel, Michiel, ed. (2001) [1994], "Schrödinger equation",
Encyclopedia of Mathematics, Springer Science+Business Media B.V. /
v t e
Background Introduction History timeline Glossary Classical mechanics Old quantum theory Fundamentals Bra–ket notation
Casimir effect
Complementarity
Density matrix
ground state excited state degenerate levels Vacuum state Zero-point energy QED vacuum QCD vacuum Hamiltonian
Operator
Qubit
Qutrit
Observable
Formulations Formulations Heisenberg Interaction Matrix mechanics Schrödinger Path integral formulation Phase space Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger Interpretations Interpretations
Bayesian
Consistent histories
Copenhagen
de Broglie–Bohm
Ensemble
Hidden variables
Many-worlds
Objective collapse
Experiments Afshar
Bell's inequality
Cold
Science
Technology
Timeline
Extensions
Axiomatic quantum field theory
Category Portal:Physics Commons Authority control |