The Hamiltonian in Quantum Mechanics

Associated with each measurable parameter in a physical system is a quantum mechanical operator, and the operator associated with the system energy is called the Hamiltonian. In classical mechanics, the system energy can be expressed as the sum of the kinetic and potential energies. For quantum mechanics, the elements of this energy expression are transformed into the corresponding quantum mechanical operators. The Hamiltonian contains the operations associated with the kinetic and potential energies and for a particle in one dimension can be written:

每个物理系统的可测量参数都对应一个量子力学算符，而与系统能量对应的算符称为哈密顿量。在经典力学中，系统的能量可以表示为动能和势能之和。对于量子力学，这个能量表达式的各项被转化为相应的量子力学算符。哈密顿量包含与动能和势能相关的操作，并且对于一维粒子可以写成：

Operating on the wavefunction with the Hamiltonian produces the Schrodinger equation. In the time independent Schrodinger equation, the operation may produce specific values for the energy called energy eigenvalues. This situation can be shown in the form

对波函数作用于哈密顿量得到薛定谔方程。在时间无关的薛定谔方程中，这种操作可能产生特定的能量值，称为能量本征值。这种情形可以表示为

其中，能量的具体值称为能量本征值，而函数Ψ_i称为本征函数。

In addition to its role in determining system energies, the Hamiltonian operator generates the time evolution of the wavefunction in the form

除了确定系统能量的作用外，哈密顿算符还以形式生成波函数的时间演化

The full role of the Hamiltonian is shown in the time dependent Shrodinger equation where both its spatial and time operations manifest themselves.

哈密顿量的全部作用在时间依赖的薛定谔方程中得到体现，其中它的空间和时间操作都表现出来。