Schrodinger Equation

The Schrodinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but given a large number of events, the Schrodinger equation will predict the distribution of results.

The kinetic and potential energies are transformed into the Hamiltonian which acts upon the wavefunction to generate the evolution of the wavefunction in time and space. The Schrodinger equation gives the quantized energies of the system and gives the form of the wavefunction so that other properties may be calculated.

Particle in a Box

The idealized situation of a particle in a box with infinitely high walls is an application of the Schrodinger equation which yields some insights into particle confinement. The wavefunction must be zero at the walls and the solution for the wavefunction yields just sine waves. 理想化情况下，一个具有无限高壁的粒子在盒中的情形是薛定谔方程的应用，它提供了一些关于粒子束缚的见解。波函数在壁处必须为零，而波函数的解仅给出正弦波。 The longest wavelength is and the higher modes have wavelengths given by When this is substituted into the DeBroglie relationship it yields momentum 最长的波长是，更高阶的模式的波长由 给出。当这个代入德布罗意关系式时，得到动量

Particle in a Box

Calculation 计算

When the momentum expression for the particle in a box is used to calculate the energy associated with the particle Though oversimplified, this indicates some important things about bound states for particles: 1. The energies are quantized and can be characterized by a quantum number n 2. The energy cannot be exactly zero. 3. The smaller the confinement, the larger the energy required. 当用于计算盒中粒子的动量表达式时，尽管过于简化，它仍能表明一些关于束缚态的重要事情：1. 能量是量子化的，并可以用量子数n来表征；2. 能量不能恰好为零；3. 粒子的束缚越弱，所需能量越大。 If a particle is confined into a rectangular volume, the same kind of process can be applied to a three-dimensional "particle in a box", and the same kind of energy contribution is made from each dimension. The energies for a three-dimensional box are 如果一个粒子被限制在一个矩形体积内，同样的过程也可以应用于三维的‘粒子在盒中’问题，每个维度都会贡献相同的能量。三维盒子的能级为 This gives a more physically realistic expression for the available energies for contained particles. This expression is used in determining the density of possible energy states for electrons in solids. 这给出了对含于其中的粒子可用能量更符合物理现实的表达式。该表达式用于确定固体中电子可能的能量态密度。

Particle in a Box Calculation

For a one-dimensional particle in a box , the particle energy for a box of dimension L can be calculated below. For a three-dimensional box there will be three values for the quantum number n. The energies for each dimension could be calculated and added. The implication of that addition is that it takes more energy to confine a particle in three dimensions than in one, and that the minimum confinement energy for a 3D box of dimension L is three times that of a 1D box.

L = x 10^ m = a₀ = fermi* = proton radii**,

and mass = x 10^ kg = m_e = m_p = MeV/c²,

then the energy for state n = for a one-dimensional box is

E = x 10^joules = eV = MeV = GeV.

The ground state of a three-dimensional box of dimension L can be obtained by setting n=1 for all three dimensions, giving an energy three times the ground state energy of the one-dimensional box. The ground state for the three-dimensional box would be 三维盒的基态可通过将三个维度的量子数n设为1来获得，其能量是一维盒基态能量的三倍。三维盒的基态为 E3D ground = x 10^joules = eV = MeV = GeV. E 3D ground = 10^3 joules = eV = MeV = GeV. 中文译文中的待填/计算数值依次对应：1：eb3 2：ep3 3：ev3 4：mev3 5：gev3。实际数值以上方原输入框为准。 * 1 fermi = 10-15m. * 1 佛米 = 10⁻¹⁵ 米。 ** Classical radius of proton calculated using the nominal nuclear density of 2.3x1017kg/m3 利用名义上的核密度2.3×10¹⁷ kg/m³计算质子的经典半径 Note: If the energy value is changed, then the state is set to n=1 and the box dimension L is calculated. Any other changes initiate a recalculation of the energy. L = x 10^ m = a 0 = fermi* = proton radii**, and mass = x 10^ kg = m e = m p = MeV/c 2 , then the energy for state n = for a one-dimensional box is E = x 10^ joules = eV = MeV = GeV. Note: If the energy value is changed, then the state is set to n=1 and the box dimension L is calculated. Any other changes initiate a recalculation of the energy.（专名或术语） 中文译文中的待填/计算数值依次对应：1：lb 2：lp 3：a 4：lf 5：pr 6：mb 7：mp 8：em 9：pm 10：mmev 11：n 12：eb 13：ep 14：ev 15：mev 16：gev。实际数值以上方原输入框为准。

Compare to uncertainty principle 与不确定性原理比较

Energies in eV 以电子伏特为单位的能

Finite wall box 有限墙盒

Example of required energy to confine particles 将粒子约束所需的能量示例