19th Ave New York, NY 95822, USA

# 物理代考| Higher Dimensions 量子力学代写

## 物理代写

3.

$3.6$ Higher Dimensions
So far, for simplicity, we have worked in just one dimension where the Schrödinger equation reads
$$\left[-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}+V(x)\right] \Psi(x, t)=i \hbar \frac{\partial \Psi(x, t)}{\partial t}$$
Here the partial derivatives imply that the other variable in the set $(x, t)$ is to be kept constant. To increase the number of dimensions, we can simply follow our work on the wave equation and replace
$$\frac{\partial^{2}}{\partial x^{2}} \rightarrow \nabla^{2}$$
where $\nabla^{2}$ is the laplacian
\begin{aligned} \nabla^{2} &=\frac{\partial^{2}}{\partial x^{2}} & & ; \text { one dimension } \ &=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}} & & ; \text { two dimensions } \ &=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}} & & ; \text { three dimensions } \end{aligned}
This is equivalent to writing the Schrödinger equation as
$$H \Psi(\vec{x}, t)=\left[\frac{\vec{p}^{2}}{2 m}+V(\vec{x})\right] \Psi(\vec{x}, t)=i \hbar \frac{\partial \Psi(\vec{x}, t)}{\partial t}$$
22
Introduction to Quantum Mechanics
and expanding the momentum to read
$$p_{j}=\frac{\hbar}{i} \frac{\partial}{\partial x_{j}} \quad ; j=1,2, \cdots$$
where the index $j$ now labels the cartesian axes.
As one example, consider a particle of mass $m$ in a square twodimensional box with sides $L$. Here the boundary conditions are those of walls, and the eigenfunctions and eigenvalues are evidently
\begin{aligned} \psi_{n_{x}, n_{y}}(x, y) &=\left(\frac{2}{L}\right) \sin \left(\frac{n_{x} \pi x}{L}\right) \sin \left(\frac{n_{y} \pi y}{L}\right) \quad ;\left(n_{x}, n_{y}\right)=1,2,3, \cdots \ E_{n_{x}, n_{y}} &=\frac{\hbar^{2} \pi^{2}}{2 m L^{2}}\left(n_{x}^{2}+n_{y}^{2}\right) \end{aligned}
The general solution to the Schrödinger equation is correspondingly
$$\Psi(x, y, t)=\sum_{n_{x}} \sum_{n_{y}} c_{n_{z}, n_{y}} \psi_{n_{x}, n_{y}}(x, y) e^{-i E_{n_{x}, n_{y}} t / \hbar}$$

## 物理代考

$3.6$ 更高的尺寸

$$\left[-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}+V(x)\right] \Psi(x, t)=i \hbar \frac{\partial \Psi(x, t)}{\partial t}$$

$$\frac{\partial^{2}}{\partial x^{2}} \rightarrow \nabla^{2}$$

$$\开始{对齐} \nabla^{2} &=\frac{\partial^{2}}{\partial x^{2}} & & ; \text { 一维 } \ &=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}} & & ; \text { 二维 } \ &=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2 }}{\partial z^{2}} & & ; \text { 三个维度 } \end{对齐}$$

$$H \Psi(\vec{x}, t)=\left[\frac{\vec{p}^{2}}{2 m}+V(\vec{x})\right] \Psi(\vec {x}, t)=i \hbar \frac{\partial \Psi(\vec{x}, t)}{\partial t}$$
22

$$p_{j}=\frac{\hbar}{i} \frac{\partial}{\partial x_{j}} \quad ; j=1,2, \cdots$$

$$\开始{对齐} \psi_{n_{x}, n_{y}}(x, y) &=\left(\frac{2}{L}\right) \sin \left(\frac{n_{x} \pi x} {L}\right) \sin \left(\frac{n_{y} \pi y}{L}\right) \quad ;\left(n_{x}, n_{y}\right)=1,2 ,3, \cdots \ E_{n_{x}, n_{y}} &=\frac{\hbar^{2} \pi^{2}}{2 m L^{2}}\left(n_{x}^{2} +n_{y}^{2}\右） \end{对齐}$$

$$\Psi(x, y, t)=\sum_{n_{x}} \sum_{n_{y}} c_{n_{z}, n_{y}} \psi_{n_{x}, n_{y} }(x, y) e^{-i E_{n_{x}, n_{y}} t / \hbar}$$

Matlab代写