Varieties of Wave Equations

Early in the 20th century, electrons were shown to have wave properties, and the wave-particle duality became a part of our understanding of nature. The mathematics for describing the behavior of such electron waves might be expected to be similar to that for describing classical waves, such as the wave on a stretched string

or a plane electromagnetic wave

The wave equation developed by Erwin Schrodinger in 1926 shows some similarities in its one-dimensional form:

The Postulates of Quantum Mechanics

	1. Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system. 任何在保守力场中运动的粒子都与一个波函数相关，该波函数决定了关于系统的所有可知信息。
	2. With every physical observable q there is associated an operator Q, which when operating upon the wavefunction associated with a definite value of that observable will yield that value times the wavefunction. 每一个物理可观测量q都对应一个算符Q，当该算符作用于与某一确定值相关的波函数时，会得到该值乘以波函数。
	3. Any operator Q associated with a physically measurable property q will be Hermitian. 任何与物理可测量属性q相关的算符Q都是厄米特的。
	4. The set of eigenfunctions of operator Q will form a complete set of linearly independent functions. 算子Q的本征函数集将构成一组线性无关的函数。
	5. For a system described by a given wavefunction, the expectation value of any property q can be found by performing the expectation value integral with respect to that wavefunction. 对于由给定波函数描述的系统，任何性质q的期望值可通过对该波函数进行期望值积分来求得。
	6. The time evolution of the wavefunction is given by the time dependent Schrodinger equation. 波函数的时间演化由时间依赖的薛定谔方程给出。

The Wavefunction Postulate

It is one of the postulates of quantum mechanics that for a physical system consisting of a particle there is an associated wavefunction. This wavefunction determines everything that can be known about the system. The wavefunction is assumed here to be a single-valued function of position and time, since that is sufficient to guarantee an unambiguous value of probability of finding the particle at a particular position and time. The wavefunction may be a complex function, since it is its product with its complex conjugate which specifies the real physical probability of finding the particle in a particular state.

Constraints on Wavefunction

In order to represent a physically observable system, the wavefunction must satisfy certain constraints:

Further discussion 进一步讨论

1. Must be a solution of the Schrodinger equation. 必须是薛定谔方程的解。 2. Must be normalizable. This implies that the wavefunction approaches zero as x approaches infinity. 必须是可归一化的。这意味着波函数在x趋向于无穷大时应趋于零。 3. Must be a continuous function of x. 必须是x的连续函数。 4. The slope of the function in x must be continuous. Specifically 具体来说 must be continuous. 必须是连续的。 x方向上的函数斜率必须连续。 These constraints are applied to the boundary conditions on the solutions, and in the process help determine the energy eigenvalues. 这些约束被应用于解的边界条件，在此过程中有助于确定能量本征值。

Probability in Quantum Mechanics

The wavefunction represents the probability amplitude for finding a particle at a given point in space at a given time. The actual probability of finding the particle is given by the product of the wavefunction with its complex conjugate (like the square of the amplitude for a complex function).

Since the probability must be = 1 for finding the particle somewhere, the wavefunction must be normalized. That is, the sum of the probabilities for all of space must be equal to one. This is expressed by the integral

Examples of normalization 归一化的例子

Normalization Examples

In order to use the wavefunction calculated from the Schrodinger equation to determine the value of any physical observable, it must be normalized so that the probability integrated over all space is equal to one.