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# 物理代写|量子力学代写Quantum mechanics代考|The Minimum-Uncertainty Wave Packet

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## 物理代写|量子力学代写Quantum mechanics代考|The Minimum-Uncertainty Wave Packet

We have twice encountered wave functions that $b i t$ the position-momentum uncertainty limit $\left(\sigma_x \sigma_p=\hbar / 2\right)$ : the ground state of the harmonic oscillator (Problem 2.11) and the Gaussian wave packet for the free particle (Problem 2.21). This raises an interesting question: What is the most general minimum-uncertainty wave packet? Looking back at the proof of the uncertainty principle, we note that there were two points at which inequalities came into the argument: Equation $\underline{3.59}$ and Equation $\underline{3.60}$. Suppose we require that each of these be an equality, and see what this tells us about $\Psi$.

The Schwarz inequality becomes an equality when one function is a multiple of the other: $g(x)=c f(x)$ , for some complex number $c$ (see Problem A.5). Meanwhile, in Equation $3.60 \mathrm{I}$ threw away the real part of $z$; equality results if $\operatorname{Re}(z)=0$, which is to say, if $\operatorname{Re}\langle f \mid g\rangle=\operatorname{Re}(c\langle f \mid f\rangle)=0$. Now, $\langle f \mid f\rangle$ is certainly real, so this means the constant $c$ must be pure imaginary-let’s call it $i a$. The necessary and sufficient condition for minimum uncertainty, then, is
$$g(x)=\operatorname{iaf}(x), \quad \text { where } a \text { is real. }$$
For the position-momentum uncertainty principle this criterion becomes:
$$\left(-i \hbar \frac{d}{d x}-\langle p\rangle\right) \Psi=i a(x-\langle x\rangle) \Psi,$$
which is a differential equation for $\Psi$ as a function of $x$. Its general solution (see Problem 3.17) is
$$\Psi(x)=A e^{-a(x-\langle x\rangle)^2 / 2 \hbar} e^{i\langle p\rangle x / \hbar} .$$

## 物理代写|量子力学代写Quantum mechanics代考|The Energy-Time Uncertainty Principle

The position-momentum uncertainty principle is often written in the form
$$\Delta x \Delta p \geq \frac{\hbar}{2}$$
$\Delta x$ (the “uncertainty” in $x$ ) is loose notation (and sloppy language) for the standard deviation of the results of repeated measurements on identically prepared systems. ${ }^{23}$ Equation $\underline{3.71}$ is often paired with the energy-time uncertainty principle,
$$\Delta t \Delta E \geq \frac{\hbar}{2}$$
Indeed, in the context of special relativity the energy-time form might be thought of as a consequence of the position-momentum version, because $x$ and $t$ (or rather, $c t$ ) go together in the position-time four-vector, while $p$ and $E$ (or rather, $E / c$ ) go together in the energy-momentum four-vector. So in a relativistic theory Equation 3.72 would be a necessary concomitant to Equation 3.71 . But we’re not doing relativistic quantum mechanics. The Schrödinger equation is explicitly nonrelativistic: It treats $t$ and $x$ on a very unequal footing (as a differential equation it is first-order in $t$, but second-order in $x$ ), and Equation 3.72 is emphatically not implied by Equation 3.71. My purpose now is to derive the energy-time uncertainty principle, and in the course of that derivation to persuade you that it is really an altogether different beast, whose superficial resemblance to the position-momentum uncertainty principle is actually quite misleading.

## 物理代写|量子力学代写Quantum mechanics代考|The Minimum-Uncertainty Wave Packet

$$g(x)=\operatorname{iaf}(x), \quad \text { where } a \text { is real. }$$

$$\left(-i \hbar \frac{d}{d x}-\langle p\rangle\right) \Psi=i a(x-\langle x\rangle) \Psi,$$

$$\Psi(x)=A e^{-a(x-\langle x\rangle)^2 / 2 \hbar} e^{i\langle p\rangle x / \hbar} .$$

## 物理代写|量子力学代写Quantum mechanics代考|The Energy-Time Uncertainty Principle

$$\Delta x \Delta p \geq \frac{\hbar}{2}$$
$\Delta x$ ($x$中的“不确定度”)是对相同制备系统的重复测量结果的标准偏差的松散符号(和草率的语言)。${ }^{23}$方程$\underline{3.71}$常与能量-时间不确定性原理配对，
$$\Delta t \Delta E \geq \frac{\hbar}{2}$$

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