Persamaan Schrödinger: Perbedaan antara revisi
Konten dihapus Konten ditambahkan
Tidak ada ringkasan suntingan |
Tidak ada ringkasan suntingan |
||
Baris 14:
<math>\mathrm{i}</math> adalah [[bilangan imaginer]], <math>t</math> adalah [[waktu]], ∂ / ∂<math>t</math> adalah [[turunan parsial]] terhadap <math>t</math>, ħ adalah [[konstanta Planck]] dibagi 2π, ψ(<math>t</math>) adalah [[fungsi gelombang]], dan H(<math>t</math>) adalah [[Hamiltonian]].
==Persamaan==
===Persamaan bergantung-waktu===
Bentuk persamaan Schrödinger tergantung dari kondisi fisiknya (lihat dibawah untuk contoh-contoh khusus). Bentuk paling umumnya adalah [[Persamaan Schrödinger#Tergantung waktu|persamaan tergantung-waktu]] yang menjelaskan sebuah sistem berkembang dengan waktu:<ref name=Shankar1994>
{{cite book
|last=Shankar |first=R.
|year=1994
|title=Principles of Quantum Mechanics
|edition=2nd
|publisher=[[Kluwer Academic]]/[[Plenum Publishers]]
|isbn=978-0-306-44790-7
}}</ref>{{rp|143}}
[[File:Wave packet (dispersion).gif|thumb|200px|Sebuah [[fungsi gelombang]] yang memenuhi persamaan Schrodinger nonrelativistik dengan {{math|''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.]]
{{Equation box 1
|indent=:
|title='''Persamaan Schrödinger tergantung-waktu''' ''(umum)''
|equation=<math>i \hbar \frac{\partial}{\partial t}\vert\Psi(\mathbf{r},t)\rangle = \hat H\vert\Psi(\mathbf{r},t)\rangle</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}
dengan {{math|''i''}} adalah [[satuan imajiner]], {{math|''ħ''}} adalah [[konstanta Planck]] tereduksi yang sama dengan :<math>\hbar = \frac{h}{2 \pi}</math>, lambang {{math|{{sfrac|∂|∂''t''}}}} menunjukkan [[turunan parsial]] terhadap [[waktu]] {{math|''t''}}, {{math|''Ψ''}} (huruf Yunani [[psi (huruf)|psi]]) adalah [[fungsi gelombang]] sistem kuantum, {{math|'''r'''}} dan {{math|''t''}} adalah posisi vektor dan waktu, dan {{math|''Ĥ''}} adalah [[operator (fisika)|operator]] [[Hamiltonian (mekanika kuantum)|Hamiltonian]] (yang mengkarakterisasi total energi sistem).
[[File:StationaryStatesAnimation.gif|300px|thumb|right|Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a [[quantum harmonic oscillator|harmonic oscillator]]. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The [[probability distribution]] of finding the particle with this wave function at a given position. The top two rows are examples of '''[[stationary state]]s''', which correspond to [[standing wave]]s. The bottom row is an example of a state which is ''not'' a stationary state. The right column illustrates why stationary states are called "stationary".]]
Contoh paling umum adalah persamaan [[mekanika kuantum relativistik|nonrelativistik]] untuk partikel tunggal yang bergerak dalam sebuah [[medan listrik]] (bukan [[medan magnet]]; lihat [[Persamaan Pauli]]):<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.html "Schrodinger equation"]. ''hyperphysics.phy-astr.gsu.edu''.</ref>
{{Equation box 1
|indent=:
|title='''Persamaan Schrödinger tergantung waktu dalam basis posisi'''<br/>''(partikel [[mekanika kuantum relativistik|nonrelativistik]] tunggal)''
|equation=<math>i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2\mu}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t)</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
dimana {{math|''μ''}} adalah "[[massa tereduksi]]" partikel, {{math|''V''}} [[energi potensial]], {{math|∇<sup>2</sup>}} adalah [[Laplasian]] (operator diferensial), dan {{math|''Ψ''}} adalah fungsi gelombang (lebih tepatnya dalam konteks ini adalah "fungsi gelombang ruang-posisi"). Dalam bahasa sederhana, persamaan ini berarti "total [[energi]] sama dengan [[energi kinetik]] ditambah [[energi potensial]]", namun dengan bentuk yang tidak umum.
Given the particular differential operators involved, this is a [[linear differential equation|linear]] [[persamaan diferensial parsial]]. It is also a [[diffusion equation]], but unlike the [[heat equation]], this one is also a wave equation given the [[imaginary unit]] present in the transient term.
The term ''"Schrödinger equation"'' can refer to both the general equation, or the specific nonrelativistic version. The general equation is indeed quite general, used throughout quantum mechanics, for everything from the [[Dirac equation]] to [[quantum field theory]], by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is a strictly classical approximation to reality and yields accurate results in many situations, but only to a certain extent (see [[relativistic quantum mechanics]] and [[relativistic quantum field theory]]).
To apply the Schrödinger equation, the Hamiltonian operator is set up for the system, accounting for the kinetic and potential energy of the particles constituting the system, then inserted into the Schrödinger equation. Hasil persamaan diferensial parsial The resulting partial differential equation is solved for the wave function, which contains information about the system.
==={{anchor|Time independent equation}}Persamaan tak tergantung-waktu===
The time-dependent Schrödinger equation described above predicts that wave functions can form [[standing wave]]s, called [[stationary state]]s (also called "orbitals", as in [[atomic orbital]]s or [[molecular orbital]]s). These states are particularly important as their individual study later simplifies the task of solving the time-dependent Schrödinger equation for ''any'' state. Stationary states can also be described by a simpler form of the Schrödinger equation, the ''time-independent Schrödinger equation'' (TISE).
{{Equation box 1
|indent=:
|title='''Persamaan Schrödinger tak tergantung-waktu''' (''umum'')
|equation=<math>\operatorname{\hat H}\vert\Psi\rangle=E\vert\Psi\rangle</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}
where ''{{math|E}}'' is a constant equal to the total energy of the system. This is only used when the [[Hamiltonian (quantum mechanics)|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 {{math|Ψ}}, and the result is proportional to the same wave function {{math|Ψ}}, then {{math|Ψ}} is a [[stationary state]], and the proportionality constant, {{math|E}}, is the energy of the state {{math|Ψ}}.''
In [[linear algebra]] terminology, this equation is an [[Eigenvalues and eigenvectors|eigenvalue equation]] and in this sense the wave function is an [[Eigenfunction|eigenfunction]] of the Hamiltonian operator.
As before, the most common manifestation is the [[relativistic quantum mechanics|nonrelativistic]] Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):
{{Equation box 1
|indent=:
|title='''Persamaan Schrödinger tak tergantung-waktu''' (''partikel tunggal nonrelativistik'')
|equation=<math>\left[ \frac{-\hbar^2}{2\mu}\nabla^2 + V(\mathbf{r}) \right] \Psi(\mathbf{r}) = E \Psi(\mathbf{r})</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
with definitions as above.
The time-independent Schrödinger equation is discussed further [[#Time_independent|below]].
== Referensi ==
| |||