# 物理代写|计算物理代写Computational Physics代考|PHYS175T

## 物理代写|计算物理代写Computational Physics代考|The Infinite Square Well

This problem is sometimes referred to as the particle in a box model. Classically the motion of the particle is governed by Newton’s equations, potential fields put forces on masses causing them to accelerate or change direction. At the quantum level, Newton’s equations are replaced by Schrödinger’s such that for a particle of mass $m$ moving through a (one-dimensional) potential $V(x)$ we have
$$-\frac{\hbar^2}{2 m} \frac{d^2 \psi}{d x^2}=E \psi(x)-V(x) \psi(x)$$
where $\hbar$ is Planck’s constant $h$ over $2 \pi, E$ is the total energy of the system, and $\psi(x)$ is the wavefunction of the system. Note that this is the time-independent version of the Schrödinger equation; timedependent versions also exist. Like the Newtonian equations, we solve Equation (4.12) for the unknown, in this case, $\psi(x)$. Although there is still some considerable debate over the nature of the wavefunction, certain observable quantities do depend on its form. For instance, the quantity $\psi^(x) \psi(x)$ describes its probability function, that is, the chance of finding the quantum particle at a particular location. More precisely the quantity $\psi^(x) \psi(x) d x$ is the probability of finding the particle in the region $x$ to $x+d x$.

## 物理代写|计算物理代写Computational Physics代考|The Finite Square Well

The finite square well is somewhat more realistic than the infinite square well. We define the potential as
$$V(x)=\left{\begin{array}{cc} V_0, & x<-a \\ 0, & -a \leq x \leq a \\ V_0, & x>a \end{array}\right.$$
Note that in this case the well is defined symmetrically about the origin of the $x$-axis, rather than having a barrier at $x=0$. We now consider the three distinct regions namely the region left of the well, the well itself, and the region right of the well. In Figure $4.5$, we label these regions as I, II, and III, respectively, and consider the implications of the potential field on the wavefunction in these three regions.

First, for particles with energy greater than the height of the well $V_0$ their wavefunctions are unbound, in other words, they can move freely, and have any energy. Interestingly, as the particle moves over the well it loses potential energy, which is transformed into kinetic cncrgy and the particle gains momentum. This shows an incrcase in the wavenumber of the wavefunction as the particle travels across the well, c.f. de Broglie (pronounced like Troy) momentum. This can also be seen in the differential equation. For a constant potential across x, Equation (4.12) has the form of a simple harmonic oscillator where the $E-V(x)$ term plays the role of the spring constant. As we go from regions I-II, the potential drops from $V_0$ to zero thus increasing the “spring constant” and the frequency of the oscillations of the particle. The opposite is true as we go from regions IIIII. (Strictly speaking, the wave is progressive rather than stationary so we should use the time-dependent version of Equation (4.12) to govern the physics of motion, though the outcome would at least be qualitatively the same. For arguments sake, you can consider the unbound wavefunctions are the bound states of an infinitely wide quantum well.)

## 物理代写|计算物理代写计算物理学代考|无限平方阱

$$-\frac{\hbar^2}{2 m} \frac{d^2 \psi}{d x^2}=E \psi(x)-V(x) \psi(x)$$
，其中$\hbar$是普朗克常数$h$除以$2 \pi, E$是系统的总能量，$\psi(x)$是系统的波函数。注意，这是Schrödinger方程的时间无关版本;与时间相关的版本也存在。就像牛顿方程一样，我们求解方程(4.12)的未知数，在这种情况下，$\psi(x)$。尽管对波函数的性质仍有相当大的争论，但某些可观测的量确实取决于它的形式。例如，数量$\psi^(x) \psi(x)$描述了它的概率函数，即在特定位置找到量子粒子的机会。更准确地说，数量$\psi^(x) \psi(x) d x$是在$x$到$x+d x$区域内找到粒子的概率

## 物理代写|计算物理代写计算物理学代考|有限平方阱

. . 物理代写|计算物理代写计算物理学代考|

$$V(x)=\left{\begin{array}{cc} V_0, & x<-a \ 0, & -a \leq x \leq a \ V_0, & x>a \end{array}\right.$$

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