Perturbation theory (quantum mechanics)

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In quantum mechanics, the theory of perturbation is a set of approximate schemes directly related to mathematical disorders to describe complex systems of quantum in simpler terms. The idea is to start with a simple system that mathematical solutions are known for, and add additional Hamiltonian "perturbing" that represents a weaker interruption to the system. If the disturbance is not too great, the various physical quantities associated with the disturbed system (eg energy levels and eigenstates) can be expressed as "corrections" to them from simple systems. This correction, which is small compared to its own number size, can be calculated using approximate methods such as asymptotic series. Complex systems can be learned based on simpler knowledge. In fact, it describes an intricate unsolved system using a simple and resolved system.

Video Perturbation theory (quantum mechanics)

Perkiraan Hamiltonian

The theory of perturbation is an important tool for describing real quantum systems, as it turns out very difficult to find the right solution for Schrödinger equations for Hamiltonians even moderate complexity. Hamiltonians that we know of right solutions, such as hydrogen atoms, quantum harmonic oscillators and particles in boxes, are too ideal to describe most systems. Using the theory of perturbation, we can use the known solution of these simple Hamiltonians to produce solutions for more complex systems.

Maps Perturbation theory (quantum mechanics)

Applying the perturbation theory

The theory of perturbation applies if the problem encountered can not be solved precisely, but can be formulated by adding the term "small" to the mathematical description of the problem that is completely solvable.

For example, by adding a perturbative electric potential to a quantum mechanical model of a hydrogen atom, a small shift in the hydrogen spectrum line caused by an electric field (the Stark effect) can be calculated. This is only an approximation because of the potential amount of Coulomb with unstable linear potential (not having a true bound status) even though the tunneling time (decay rate) is very long. This instability arises as an expansion of the energy spectrum lines, to which the interference theory fails to reproduce completely.

The expression generated by perturbation theory is not appropriate, but they can lead to accurate results during expansion parameters, say ? , very small. Usually, the results are expressed in the form of limited power circuits in ? that seems to blend in with the right value when summed to a higher order. After a certain order n ~ 1/? however, the result is getting worse because the series is usually different (being asymptotic series). There is a way to convert it into convergent series, which can be evaluated for large expansion parameters, most efficiently by variational methods.

In the theory of quantum electrodynamics (QED), in which electron-photon interactions are treated perturbatively, calculations of electron magnetic moments have been found to agree with experiments for eleven decimal places. In QED and other quantum field theories, special calculation techniques known as Feynman diagrams are used to systematically sum the term power series.

Limitations

Great annoyance

In some circumstances, the disturbance theory is an invalid approach to take. This happens when the system we want to describe can not be explained by the small annoyances imposed on some simple systems. In quantum chromodynamics, for example, the quark interaction with the gluon field can not be treated perturbatively at low energy because the coupling constant (expansion parameter) becomes too large.

Non-adiabatic conditions

The theory of perturbation also fails to describe the adiabatically unexpired circumstances of the "free model", including bound states and collective phenomena such as solitons. Imagine, for example, that we have a free particle system (ie not interacting), in which interesting interactions are introduced. Depending on the shape of the interaction, this can create an entirely new set of eigenstates that match the groups of particles tied to each other. An example of this phenomenon can be found in conventional superconductivity, where the mediation-phonon attraction between the conducting electrons leads to the formation of correlated electron pairs known as Cooper pairs. When confronted with such a system, one usually turns to another approximation scheme, such as the variational method and the WKB approach. This is because there is no analogue of the bonded particles in the unaffected model and the energy of a soliton usually acts as the inverse of the expansion parameter. However, if we "integrate" above the solitonic phenomenon, the nonperturbative correction in this case would be very small; the exp sequence (-1/ g ) or exp (-1/ g ² ) in the perturbation parameter g . The theory of perturbation can only detect "close" solutions with uninterrupted solutions, even if there are other solutions for invalid perturbative extensions.

Difficult computing

The problem of non-intruder systems has been somewhat mitigated by the advent of modern computers. It has become practical to get numerical non-perturbative solutions to certain problems, using methods such as functional density theory. These advances have been particularly useful for the field of quantum chemistry. Computers have also been used to perform impaired theoretical calculations to an unusually high degree of precision, which has proved important in particle physics to produce theoretical results that can be compared with experiments.

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Time-independent perturbation theory

Independent time perturbation theory is one of two categories of interference theory, the other is time-dependent disorder (see next section). In the theory of perturbation time-free the Hamiltonian disorder is static (ie, has no time dependence). Time-independent perturbation theory was presented by Erwin SchrÃÆ'¶dinger in a 1926 paper, shortly after he produced his theory in wave mechanics. In this paper Schrödinger refers to the previous work of Lord Rayleigh, which investigates the harmonic vibrations of strings that are disturbed by small inhomogeneities. This is why the theory of this disorder is often referred to as Rayleigh-SchrÃÆ'¶dinger perturbation theory .

First order correction

Kami mulai dengan Hamiltonian yang tidak terganggu H ₀ , yang juga diasumsikan tidak memiliki ketergantungan waktu. Ia telah mengetahui tingkat energi dan eigenstates, yang timbul dari persamaan Schrö¶dinger yang independen waktu:

${\ displaystyle H_ {0} \ left | n ^ {(0)} \ right \ rangle = E_ {n} ^ {(0)} \ left | n ^ { (0)} \ right \ rangle, \ qquad n = 1,2,3, \ cdots}  ÂÂ$

For simplicity, we assume that energy is separate. Superscript (0) indicates that this quantity is associated with an undisturbed system. Note the use of bra-ket notation.

We are now introducing an annoyance to the Hamiltonian. Let V be a Hamiltonian representing a weak physical disorder, such as the potential energy generated by the external field. (Thus, V is formally a Hermitian operator.) Leave ? becomes a dimensionless parameter that can take values ​​ranging from 0 (without interruption) to 1 (full interruption). The annoyed Hamiltonian

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂHÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ? Â Â Â Â Â Â Â Â V Â Â Â Â Â Â Â Â Â Â Â Â Â Â {\ displaystyle H = H_ {0} \ lambda V} Â Â}

Tingkat empowers Dan eigenstates to give Hamiltonian Terganggu Lithuanian Diberikan oleh Persama Schrödinger:

${\ displaystyle \ left (H_ {0} \ lambda V \ right) | n \ rangle = E_ {n} | n \ rangle.} Â Â$

When k = 0 , this reduces the undisturbed value, which is the first term in each series. Because the disorder is weak, energy levels and eigenstates should not deviate too much from their undisturbed values, and the terms must rapidly become smaller as we move towards higher order.

Mengganti perluasan seri daya ke persamaan Schrödinger, kita dapatkan

${\ displaystyle \ left (H_ {0} \ lambda V \ right) \ left (\ left | n ^ {(0)} \ right \ rangle \ lambda \ left | n ^ {(1)} \ right \ rangle \ cdots \ right) = \ kiri (E_ {n} ^ {(0)} \ lambda E_ {n} ^ {(1)} \ cdots \ kanan) \ kiri (\ kiri | n ^ {(0)} \ right \ rangle \ lambda \ left | n ^ {(1)} \ right \ rangle \ cdots \ right)}  ÂÂ$

Memperluas persamaan ini dan membandingkan koefisien dari setiap kekuatan ? menghasilkan serangkaian persamaan simultan yang tak terbatas. Persamaan zeroth-order hanyalah persamaan Schrödinger untuk sistem yang tidak terganggu. Persamaan orde pertama adalah

${\ displaystyle H_ {0} \ left | n ^ {(1)} \ right \ rangle V \ left | n ^ {(0)} \ right \ rangle = E_ {n} ^ {(0)} \ left | n ^ {(1)} \ right \ rangle E_ {n} ^ {(1)} \ left | n ^ {(0)} \ right \ rangle}  ÂÂ$

Operasi melalui oleh ${\ displaystyle \ langle n ^ {(0)} |} Â$ , istilah pertama di sisi kiri membatalkan istilah pertama di sisi kanan. (Ingat, Hamiltonian yang tidak terganggu adalah Hermitian). Ini mengarah, every one of them would gain energy:

${\ displaystyle E_ {n} ^ {(1)} = \ kiri \ langle n ^ {(0)} \ right | V \ left | n ^ {(0)} \ right \ rangle} Â Â$

Ini hanyalah nilai harapan dari gangguan Hamiltonian sementara sistem berada dalam keadaan tidak terganggu. Hasil ini dapat ditafsirkan dengan cara berikut: anggap gangguan tersebut diterapkan, tetapi kami menjaga sistem dalam status kuantum ${\ displaystyle | n ^ {(0)} \ rangle} Â Â$ , yang merupakan status kuantum yang valid meskipun tidak lagi merupakan eigenstate energi. Gangguan ini menyebabkan energi rata-rata dari keadaan ini meningkat dengan ${\ displaystyle \ langle n ^ {(0)} | V | n ^ {(0)} \ rangle} Â Â$ . Namun, pergeseran energi yang sebenarnya sedikit berbeda, karena eigenstate yang terganggu tidak persis sama dengan ${\ displaystyle | n ^ {(0)} \ rangle} Â Â$ . They would read Lebih lanjut ini diberikan oleh koreksi urutan kedua dan lebih tinggi that energi.

Sebelum kita menghitung koreksi terhadap eigenstate energi, kita perlu mengatasi masalah normalisasi. Kami mungkin mengira

${\ displaystyle \ left \ langle n ^ {(0)} \ right | \ left.n ^ {(0)} \ right \ rangle = 1,} Â Â$

Karena phase keseluruhan tidak ditentukan dalam mechanika kuantum, tanpa kehilangan keumuman, kita dapat menganggap ${\ displaystyle \ langle n ^ {(0)} | n ^ {(1)} \ rangle} Â$ benar-benar nyata. Karena itu,

${\ displaystyle \ left \ langle n ^ {(0)} \ right | \ left.n ^ {(1)} \ right \ rangle = - \ left \ langle n ^ {(1)} \ right | \ left.n ^ {(0)} \ right \ rangle,} Â Â$

dan kami menyimpulkan

${\ displaystyle \ left \ langle n ^ {(0)} \ right | \ left.n ^ {(1)} \ right \ rangle = 0.} Â Â$

di mana ${\ displaystyle | k ^ {(0)} \ rangle} Â Â$ berada dalam pelengkap ortogonal ${\ displaystyle | n ^ {(0)} \ rangle} Â Â$ . Persuasan party members denounce democratic dapat dyspresian sebagai

${\ displaystyle \ left (E_ {n} ^ {(0)} - H_ {0} \ right) \ left | n ^ {(1)} \ right \ rangle = \ jumlah {k \ neq n} \ kiri | k ^ {(0)} \ right \ rangle \ left \ langle k ^ {(0)} \ right | V \ left | n ^ {(0)} \ right \ rangle} Â Â$

Untuk saat ini, anggaplah bahwa tingkat energi urutan-nol tidak merosot, yaitu tidak ada eigenstat H ₀ dalam komplemen ortogonal dari ${\ displaystyle | n ^ {(0)} \ rangle} Â$ denote energy ${\ displaystyle E_ {n} ^ {(0)}} Â Â$ . Setelah mengganti nama indeks dummy penjumlahan di atas sebagai ${\ displaystyle k '} Â Â$ , kita dapat memilih ${\ displaystyle k \ neq n} Â$ , dan kalikan melalui oleh