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# 物理代考| Quantization of the Oscillator 量子力学代写

## 物理代写

6.7 Quantization of the Oscillator
We will spend some time with the quantization of the simple harmonic oscillator, since that will be central to what we do in quantum mechanics. The hamiltonian for the one-dimensional oscillator is
$$H=\frac{p^{2}}{2 m}+\frac{\kappa q^{2}}{2} \quad ; \text { oscillator }$$
Introduce the destruction and creation operators by
\begin{aligned} a &=\frac{p}{(2 m \hbar \omega)^{1 / 2}}-i q\left(\frac{m \omega}{2 \hbar}\right)^{1 / 2} \quad ; \omega^{2}=\frac{\kappa}{m} \ a^{\dagger} &=\frac{p}{(2 m \hbar \omega)^{1 / 2}}+i q\left(\frac{m \omega}{2 \hbar}\right)^{1 / 2} \end{aligned}
From our discussion of the hermiticity of operators, it is clear that $a^{\dagger}$ is the hermitian adjoint of $a .{ }^{9}$ The canonical commutation relation between the momentum and coordinate is
$$[p, q]=\frac{\hbar}{i}$$
It follows that the creation and destruction operators satisfy
$$\left[a, a^{\dagger}\right]=1 \quad \text {; commutation relation (6.48) }$$
Written in terms of these new operators, the hamiltonian takes the form
$$H=\hbar \omega\left(a^{\dagger} a+\frac{1}{2}\right) \equiv \hbar \omega\left(N+\frac{1}{2}\right)$$
${ }^{8}$ Recall $c^{2}=1 / \varepsilon_{0} \mu_{0} .$ ${ }^{9}$ See Prob. $6.4$. Here we label the coordinate more generally as $q .$
Here we have defined the number operator by $$N \equiv a^{\dagger} a \quad ; \text { number operat }$$
It fiuther follows from our discussion of hermiticity that the number oper-
Now we could write and solve the simple harmonic oscillator as a onedimensional differential Schrödinger equation in coordinate space. ${ }^{11}$ It is of great interest, however, to see how far we can get on the spectrum of the number operator, and on the application of the creation and and destruction operators to those eigenstates, using only the general principles of quantum destructions operators.

We shall therefore proceed to work in an abstract occupation number space, where we write the eigenvalue equation for the number operator in abstract form as ${ }^{12}$
$N|n\rangle=n|n\rangle \quad ;$ abstract eigenvalue equation (6.51)
The abstract eigenstates are orthonormal, with an inner product satisfying

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## 物理代考

6.7 振荡器的量化

$$H=\frac{p^{2}}{2 m}+\frac{\kappa q^{2}}{2} \quad ; \text { 振荡器 }$$

$$\开始{对齐} a &=\frac{p}{(2 m \hbar \omega)^{1 / 2}}-iq\left(\frac{m \omega}{2 \hbar}\right)^{1 / 2} \四; \omega^{2}=\frac{\kappa}{m} \ a^{\dagger} &=\frac{p}{(2 m \hbar \omega)^{1 / 2}}+iq\left(\frac{m \omega}{2 \hbar}\right)^ {1 / 2} \end{对齐}$$

$$[p, q]=\frac{\hbar}{i}$$

$$\left[a, a^{\dagger}\right]=1 \quad \text {;交换关系 (6.48) }$$

$$H=\hbar \omega\left(a^{\dagger} a+\frac{1}{2}\right) \equiv \hbar \omega\left(N+\frac{1}{2}\right)$$
${ }^{8}$ 回忆 $c^{2}=1 / \varepsilon_{0} \mu_{0} .$ ${ }^{9}$ 见概率。 6.4 美元。在这里，我们更一般地将坐标标记为 $q .$

$N|n\rangle=n|n\rangle \quad ;$ 抽象特征值方程 (6.51)

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