量子哈密頓算符
量子力學的數學基礎條目中介紹過，某體系的物理狀態可以由一個抽象希爾伯特空間中的射線來表示。或者，在研究對象是系綜時，物理狀態可用一個可計數的以機率為權重的向量序列來表示。物理上的可觀測量則由作用於該希爾伯特空間的自伴算符來描述。例如，一個自鏇自由度為1/2的粒子對應的希爾伯特空間為 C2，一個沿某直線自由運動的粒子對應的希爾伯特空間則為Lp空間，L2(R)，the space of complex-valued and square-integrable (with respect to the Lebesgue measure) functions on the real line.
量子哈密頓算符 H 是對應於體系總能的可觀測量。用數學語言來說，它是一個稠定自伴算符。如果態空間是有限空間，那么該哈密頓算符自然也是有界算符。如果態空間是無限空間，該哈密頓算符則通常無界，所以並未定義於整個空間。
物理學入門資料中常可找到下面這條公理：
哈密頓算符 H 的本徵右矢(本徵向量)，寫作 (使用狄拉克左矢-右矢符號)，提供了一組希爾伯特空間的正交歸一化的基。體系允許的能級值譜由解下列方程得到的一集本徵值給出，記作 {Ea}，
。
由於哈密頓算符 H 是厄米算符，其本徵值，即能量值總是實數。
取決於體系的希爾伯特空間的性質，能量值譜可以是離散的，也可以是連續的。事實上，還有的體系在某個能量範圍內值譜是連續的，而在另一個範圍內則是離散的。有限勢井就是一個這樣的例子，兼有離散的能量為負值的態和連續的能量為正值的態。
哈密頓算符產生量子態的時間演化。如果體系在時間t時的態是 ，那么
，
式中是h-bar。這個方程稱為薛丁格方程（它和哈密頓-雅克比方程有著相同的形式,這也是H也被稱為哈密頓算符的原因之一）。給定處於初始時間(t = 0)的初始態，我們可以積分上式得到處於任何時刻的態。特別地，當 H 與時間無關時, 那么
，
式中右邊的指數算符以通常的級數定義。可以證明它是一個么正算符，也是時間演化算法的一種常見形式（亦稱為傳播子）。
[編輯]Energy eigenket degeneracy, symmetry, and conservation laws
In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. A wave propagating in the x direction is a different state from one propagating in the y direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be degenerate.
It turns out that degeneracy occurs whenever a nontrivial unitary operator U commutes with the Hamiltonian. To see this, suppose that |a> is an energy eigenket. Then U|a> is an energy eigenket with the same eigenvalue, since
Since U is nontrivial, at least one pair of and must represent distinct states. Therefore, H has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.
The existence of a symmetry operator implies the existence of a conserved observable. Let G be the Hermitian generator of U:
U = I − iεG + O(ε2)
It is straightforward to show that if U commutes with H, then so does G:
[H,G] = 0
Therefore,
In obtaining this result, we have used the Schrödinger equation, as well as its dual,
.
Thus, the expected value of the observable G is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum.
[編輯]哈密頓等式
經典的哈密頓等式 哈密頓力學 和量子力學的哈密頓等式有著直接的類似.假設有一系列並不必要為本證態的基態 ，由這樣的類似性，我們假設這些值是離散並且是正規的， i.e.,
注意到這些基態都假設與時間無關，因而我們可以也假設哈密頓運算元也獨立於時間。
在時間t時，系統的暫時態, 可以用以上的基態展開：
這裡
The coefficients an(t) are complex variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole.
The expectation value of the Hamiltonian of this state, which is also the mean energy, is
where the last step was obtained by expanding in terms of the basis states.
Each of the an(t)'s actually corresponds to two independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use an(t) and its complex conjugate an*(t). With this choice of independent variables, we can calculate the partial derivative
By applying Schrödinger's equation and using the orthonormality of the basis states, this further reduces to
similarly, one can show that
If we define "conjugate momentum" variables πn by
then the above equations become
which is precisely the form of Hamilton's equations, with the ans as the generalized coordinates, the πns as the conjugate momenta, and taking the place of the classical Hamiltonian.wp:Hamiltonian_(quantum_mechanics)
取自"http://chemwiki.net/index.php/%E5%93%88%E5%AF%86%E9%A0%93%E7%AE%97%E7%AC%A6"
