Quantum vibration
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量子調和振動子 は、 古典調和振動子量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最も量子力学における重要なモデル系。さらに、これは正確な 解析解法が知られている数少ない量子力学系の1つである。 author=Griffiths, David J. , title=量子力学入門 , エディション=2nd , 出版社=プレンティス・ホール , 年=2004 , isbn=978-0-13-805326-0 , author-link=David Griffiths (物理学者) , URL アクセス = 登録 , url=https://archive.org/details/introductiontoel00grif_0


One-dimensional harmonic oscillator


Hamiltonian and energy eigenstates

粒子の ハミルトニアン は次のとおりです。 \hat H = \frac + \frac k ^2 = \frac + \frac m \omega^2 ^2 \, , ここで、 は粒子の質量、 は力定数、\omega = \sqrt動子の_[角周波数\hat_は_位置演算子.html" ;"title="周波数.html" ;"title="動子の [角周波数">動子の [角周波数、\hat は 位置演算子">周波数.html" ;"title="動子の [角周波数">動子の [角周波数、\hat は 位置演算子 (座標ベースで によって与えられる)、および \ hat は 運動量演算子 (座標ベースで \hat p = -i \hbar \, \partial / \partial x で与えられる) です。 フックの法則 のように、ハミルトニアンの最初の項は粒子の運動エネルギーを表し、2 番目の項はそのポテンシャル エネルギーを表します。 時間に依存しない シュレーディンガー方程式 を書くことができます。 \hat H \left, \psi \right\rangle = E \left, \psi \right\rangle ~, ここで、 は、時間に依存しない エネルギー レベル または 固有値 を指定する決定される実数を表し、解 は、その準位のエネルギー 固有状態 を示します。 波動関数スペクトル法を使用。解決策のファミリーがあることがわかりました。この基準では、 エルミート関数 \psi_n(x) = \frac \left(\frac\right )^ e^ H_n\left(\sqrt x \right), \qquad n = 0,1,2 ,\lドット. The functions ''Hn'' are the physicists' Hermite polynomials, H_n(z)=(-1)^n~ e^\frac\left(e^\right). The corresponding energy levels are E_n = \hbar \omega\bigl(n + \tfrac\bigr)=(2 n + 1) \omega~. このエネルギー スペクトルは、3 つの理由で注目に値します。まず、エネルギーが量子化されます。つまり、離散的なエネルギー値 ( の整数プラス半分の倍数) のみが可能です。これは、粒子が閉じ込められているときの量子力学系の一般的な特徴です。第二に、原子の ボーア模型箱の中の粒子とは異なり、これらの離散エネルギー準位は等間隔である。第三に、達成可能な最低エネルギー ( 基底状態 と呼ばれる 状態のエネルギー) は、ポテンシャル井戸の最小値と等しくありませんが、;これは ゼロ点エネルギーと呼ばれます。ゼロ点エネルギーのため、基底状態の振動子の位置と運動量は (古典的な振動子のように) 固定されていませんが、 ハイゼンベルグの不確定性原理に従って、小さな範囲の分散があります。 。 基底状態の確率密度は原点に集中しています。これは、エネルギーがほとんどない状態で予想されるように、粒子がほとんどの時間をポテンシャル井戸の底で過ごすことを意味します。エネルギーが増加すると、確率密度は、状態のエネルギーがポテンシャル エネルギーと一致する古典的な「ターニング ポイント」でピークに達します。 (高度に励起された状態については、以下の説明を参照してください。)これは、粒子が移動している転換点の近くでより多くの時間を費やす (したがって、発見される可能性が高い) 古典的な調和振動子と一致しています。最も遅い。したがって、 対応原理 は満たされます。さらに、 コヒーレント状態 と呼ばれる最小の不確実性を持つ特別な非分散 波束 は、図に示されているように、古典的なオブジェクトと非常によく似て振動します。それらはハミルトニアンの固有状態ではありません。


Ladder operator method

The "
ladder operator In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
" method, developed by
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
, allows extraction of the energy eigenvalues without directly solving the differential equation. It is generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators and its
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
, \begin a &=\sqrt \left(\hat x + \hat p \right) \\ a^\dagger &=\sqrt \left(\hat x - \hat p \right) \endNote these operators classically are exactly the generators of normalized rotation in the phase space of x and m\frac, ''i.e'' they describe the forwards and backwards evolution in time of a classical harmonic oscillator. These operators lead to the useful representation of \hat and \hat, \begin \hat x &= \sqrt(a^\dagger + a) \\ \hat p &= i\sqrt(a^\dagger - a) ~. \end The operator is not
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
, since itself and its adjoint are not equal. The energy eigenstates (also known as Fock states), when operated on by these ladder operators, give \begin a^\dagger, n\rangle &= \sqrt , n + 1\rangle \\ a, n\rangle &= \sqrt , n - 1\rangle. \end It is then evident that , in essence, appends a single quantum of energy to the oscillator, while removes a quantum. For this reason, they are sometimes referred to as "creation" and "annihilation" operators. From the relations above, we can also define a number operator , which has the following property: \begin N &= a^\dagger a \\ N\left, n \right\rangle &= n\left, n \right\rangle. \end The following commutators can be easily obtained by substituting the
canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...
, , a^\dagger= 1,\qquad , a^\dagger= a^,\qquad , a= -a, And the Hamilton operator can be expressed as \hat H = \hbar\omega\left(N + \frac\right), so the eigenstate of is also the eigenstate of energy. The commutation property yields \begin Na^, n\rangle &= \left(a^\dagger N + , a^\daggerright), n\rangle \\ &= \left(a^\dagger N + a^\dagger\right), n\rangle \\ &= (n + 1)a^\dagger, n\rangle, \end and similarly, Na, n\rangle = (n - 1)a , n \rangle. This means that acts on to produce, up to a multiplicative constant, , and acts on to produce . For this reason, is called a annihilation operator ("lowering operator"), and a creation operator ("raising operator"). The two operators together are called
ladder operator In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
s. In quantum field theory, and are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy. Given any energy eigenstate, we can act on it with the lowering operator, , to produce another eigenstate with less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to . However, since n = \langle n , N , n \rangle = \langle n , a^\dagger a , n \rangle = \Bigl(a , n \rangle \Bigr)^\dagger a , n \rangle \geqslant 0, the smallest eigen-number is 0, and a \left, 0 \right\rangle = 0. In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstates. Furthermore, we have shown above that \hat H \left, 0\right\rangle = \frac \left, 0\right\rangle Finally, by acting on , 0⟩ with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates \left\, such that \hat H \left, n \right\rangle = \hbar\omega \left( n + \frac \right) \left, n \right\rangle, which matches the energy spectrum given in the preceding section. Arbitrary eigenstates can be expressed in terms of , 0⟩, , n\rangle = \frac , 0\rangle.


Analytical questions

前述の分析は代数的であり、上げ演算子と下げ演算子の間の交換関係のみを使用しています。代数分析が完了したら、分析的な問題に取り掛かる必要があります。まず、基底状態、つまり方程式 a\psi_0 = 0 の解を見つける必要があります。位置表現では、これは一次微分方程式です。 \left(x+\frac\frac\right)\psi_0 = 0, その解はガウス分布であることが容易にわかります In the
continuum limit In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processe ...
, , , while is held fixed. The canonical coordinates devolve to the decoupled momentum modes of a scalar field, \phi_k, whilst the location index (''not the displacement dynamical variable'') becomes the parameter argument of the scalar field, \phi (x,t).


Molecular vibrations

* The vibrations of a
diatomic molecule Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. O ...
are an example of a two-body version of the quantum harmonic oscillator. In this case, the angular frequency is given by \omega = \sqrt where \mu = \frac is the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
and m_1 and m_2 are the masses of the two atoms. * The Hooke's atom is a simple model of the
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
atom using the quantum harmonic oscillator. * Modelling phonons, as discussed above. * A charge q with mass m in a uniform magnetic field \mathbf is an example of a one-dimensional quantum harmonic oscillator:
Landau quantization In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, call ...
.


See also

* Quantum pendulum *
Quantum machine A quantum machine is a human-made device whose collective motion follows the laws of quantum mechanics. The idea that macroscopic objects may follow the laws of quantum mechanics dates back to the advent of quantum mechanics in the early 20th ce ...
* Gas in a harmonic trap *
Creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
*
Coherent state In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harm ...
*
Morse potential The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the qua ...
*
Bertrand's theorem In classical mechanics, Bertrand's theorem states that among central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits. The f ...
*
Mehler kernel The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator. Mehler's formula defined a function and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials (.) ba ...
* Molecular vibration


References


External links


Quantum Harmonic Oscillator Live 3D intensity plots of quantum harmonic oscillatorDriven and damped quantum harmonic oscillator (lecture notes of course "quantum optics in electric circuits")
{{DEFAULTSORT:Quantum Harmonic Oscillator Quantum models Oscillators