UA OPTI570 量子力学22 2-D Isotropic Q.H.O.简介

基本概念
2-D state的矩阵表示

基本概念

这一讲介绍一个2-D量子谐振子的例子作为量子谐振子这部分的结尾。考虑2-D Isotropic Q.H.O.，
V=Vx+Vy=12mw2(X2+Y2)H=Hx+Hy=Px2+Py22m+12mw2(X2+Y2)V=V_x+V_y=\frac{1}{2}mw^2(X^2+Y^2) \\ H = H_x+H_y=\frac{P_x^2+P_y^2}{2m}+\frac{1}{2}mw^2(X^2+Y^2)V=Vx+Vy=21mw2(X2+Y2)H=Hx+Hy=2mPx2+Py2+21mw2(X2+Y2)

定义湮灭算符
ax=12(Xσ+iσPxℏ)ay=12(Yσ+iσPyℏ)a_x=\frac{1}{\sqrt{2}}\left( \frac{X}{\sigma}+\frac{i\sigma P_x}{\hbar}\right) \\ a_y=\frac{1}{\sqrt{2}}\left( \frac{Y}{\sigma}+\frac{i\sigma P_y}{\hbar}\right)ax=21(σX+ℏiσPx)ay=21(σY+ℏiσPy)

Coherent state满足
ax∣αx⟩=αx∣αx⟩,αx=12(⟨X⟩σ+iσ⟨Px⟩ℏ),∣αx⟩=e−∣αx∣22∑nx=0+∞αxnxnx!∣nx⟩ay∣αy⟩=αy∣αy⟩,αy=12(⟨Y⟩σ+iσ⟨Py⟩ℏ),∣αy⟩=e−∣αy∣22∑ny=0+∞αynyny!∣ny⟩a_x |\alpha_x \rangle = \alpha_x | \alpha_x \rangle,\alpha_x=\frac{1}{\sqrt{2}}\left( \frac{\langle X \rangle }{\sigma}+\frac{i\sigma \langle P_x \rangle }{\hbar}\right),|\alpha_x \rangle = e^{-\frac{|\alpha_x|^2}{2}}\sum_{n_x=0}^{+\infty} \frac{\alpha_x^{n_x}}{\sqrt{n_x!}}|n_x \rangle \\ a_y |\alpha_y \rangle = \alpha_y | \alpha_y\rangle,\alpha_y=\frac{1}{\sqrt{2}}\left( \frac{\langle Y \rangle }{\sigma}+\frac{i\sigma \langle P_y \rangle }{\hbar}\right),|\alpha_y \rangle = e^{-\frac{|\alpha_y|^2}{2}}\sum_{n_y=0}^{+\infty} \frac{\alpha_y^{n_y}}{\sqrt{n_y!}}|n_y \rangleax∣αx⟩=αx∣αx⟩,αx=21(σ⟨X⟩+ℏiσ⟨Px⟩),∣αx⟩=e−2∣αx∣2nx=0∑+∞nx!αxnx∣nx⟩ay∣αy⟩=αy∣αy⟩,αy=21(σ⟨Y⟩+ℏiσ⟨Py⟩),∣αy⟩=e−2∣αy∣2ny=0∑+∞ny!αyny∣ny⟩

记x与y方向的energy eigenstate分别为Ex={∣nx⟩}\mathcal{E}_x=\{|n_x \rangle\}Ex={∣nx⟩}与Ey={∣ny⟩}\mathcal{E}_{y}=\{|n_y \rangle\}Ey={∣ny⟩}，则系统的energy eigenstate为
E={∣nx,ny⟩}=Ex⊗Ey\mathcal{E} =\{|n_x,n_y \rangle\}= \mathcal{E}_x \otimes \mathcal{E}_yE={∣nx,ny⟩}=Ex⊗Ey

其中⊗\otimes⊗表示张量积。所以
∣αx,αy⟩=e−∣αx∣22e−∣αy∣22∑nx,ny=0+∞αxnxαynynx!ny!∣nx,ny⟩|\alpha_x,\alpha_y \rangle = e^{-\frac{|\alpha_x|^2}{2}}e^{-\frac{|\alpha_y|^2}{2}}\sum_{n_x,n_y=0}^{+\infty} \frac{\alpha_x^{n_x}\alpha_y^{n_y}}{\sqrt{n_x!}\sqrt{n_y!}}|n_x,n_y \rangle∣αx,αy⟩=e−2∣αx∣2e−2∣αy∣2nx,ny=0∑+∞nx!ny!αxnxαyny∣nx,ny⟩

湮灭算符的作用为
ax∣αx,αy⟩=αx∣αx,αy⟩axay∣αx,αy⟩=αx∣αx,αy⟩axay†∣nx,ny⟩=nxny+1∣nx−1,ny+1⟩,nx≥1a_x|\alpha_x,\alpha_y \rangle=\alpha_x | \alpha_x,\alpha_y \rangle \\ a_xa_y|\alpha_x,\alpha_y \rangle=\alpha_x | \alpha_x,\alpha_y \rangle \\ a_xa_y^{\dag}|n_x,n_y \rangle=\sqrt{n_x}\sqrt{n_y+1} | n_x-1,n_y+1 \rangle,n_x \ge 1ax∣αx,αy⟩=αx∣αx,αy⟩axay∣αx,αy⟩=αx∣αx,αy⟩axay†∣nx,ny⟩=nxny+1∣nx−1,ny+1⟩,nx≥1

Displacement Operator的作用为
Dx(αx)Dy(αy)∣nx=0,ny=0⟩=∣αx,αy⟩D_x(\alpha_x)D_y(\alpha_y)|n_x=0,n_y=0 \rangle = |\alpha_x,\alpha_y \rangleDx(αx)Dy(αy)∣nx=0,ny=0⟩=∣αx,αy⟩

2-D state的矩阵表示

按先y后x的顺序，比如nx,ny=0,1,2n_x,n_y=0,1,2nx,ny=0,1,2，
E={∣00⟩,∣01⟩,∣10⟩,∣02⟩,∣11⟩,∣20⟩,∣12⟩,∣21⟩,∣22⟩}\mathcal{E}=\{|00 \rangle,|01 \rangle,|10 \rangle,|02 \rangle,|11 \rangle,|20 \rangle,|12 \rangle,|21 \rangle,|22\rangle\}E={∣00⟩,∣01⟩,∣10⟩,∣02⟩,∣11⟩,∣20⟩,∣12⟩,∣21⟩,∣22⟩}

这几个状态的向量表示为e1,e2,⋯,e8e_1,e_2,\cdots,e_8e1,e2,⋯,e8，哈密顿量的矩阵表示为
ℏw⋅diag(1,2,2,3,3,3,4,4,5)\hbar w \cdot diag(1,2,2,3,3,3,4,4,5)ℏw⋅diag(1,2,2,3,3,3,4,4,5)

如果某个量子态满足
∣ψ⟩=∑nx,nycnx,ny∣nx,ny⟩|\psi \rangle = \sum_{n_x,n_y}c_{n_x,n_y}|n_x,n_y \rangle∣ψ⟩=nx,ny∑cnx,ny∣nx,ny⟩