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# 物理代考| Stern–Gerlach Experiment 量子力学代写

## 物理代写

8.1 Stern-Gerlach Experiment
In this experiment a beam of neutral particles with internal angular momentum $\hbar \overrightarrow{\mathcal{S}}$, here assumed to be spin-1/2 with $\mathcal{S}_{z}=\pm 1 / 2$, and a magnetic moment $\vec{\mu}=2 \mu \overrightarrow{\mathcal{S}}$ in the direction of the spin, is passed through an inhomogeneous magnetic field (see Fig. 8.1).

Fig. $8.1$ Stern-Gerlach experiment on a spin-1/2 system with a magnetic moment $\vec{\mu}=$ $2 \mu \overrightarrow{\mathcal{S}}$, and $\mathcal{S}{\Sigma}=\pm 1 / 2$. The $z$-axis is in the plane and vertical. 67 68 Introduction to Quantum Mechanics The particles feel a force in the $z$-direction of $$F{z}=\mu_{z} \frac{d B_{z}}{d z}$$
Instead of seeing a continuous distribution of particles coming out of the detector in the $z$-direction, one observes only two beams, corresponding to $\mathcal{S}{z}=\pm 1 / 2$. This illustrates the discrete quantization of the angular momentum. One observes just the eigenvalues of $\mathcal{S}{z}$.

In quantum mechanics we understand what is happening by saying the initial internal state of each particle is a linear combination of the two spin states ${ }^{1}$

The probability that we will measure spin up is then $\left|c_{\uparrow}(t)\right|^{2}$, and the probability that we will measure spin down is $\left|c_{\downarrow}(t)\right|^{2}$, where
$$\left|c_{\uparrow}(t)\right|^{2}+\left|c_{\downarrow}(t)\right|^{2}=1$$
The internal Schrödinger equation tracks the behavior of both components as time progresses.

Suppose one now passes the top beam through a second detector identical to the first one as illustrated in Fig. 8.2.
Fig. $8.2$ Repeat of Stern-Gerlach experiment on upper beam using detector (A).
One will now observe that all of the particles coming out again have their spin up, and there are none coming out with their spin down. ${ }^{2}$ We conclude from this that measurements are reproducible, and if we measure that the particle has spin up, then another measurement immediately afterwards will again say that it has spin up.
${ }^{1}$ Here the states $|\uparrow\rangle$ and $|\downarrow\rangle$ are one-particle states in the abstract occupation number
space. 2We assume pure pass measurements here.
Quantum Measurements
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But in quantum mechanics, we actually have something more profound. The act of the first measurement has changed the system. It is no longer in the state in Eq. (8.2). The act of measurement has reduced the basis. The act of measurement has placed it entirely in the new state
$$\left|\psi_{\text {int }}(t)\right\rangle=c(t)|\uparrow\rangle \quad ;|c(t)|^{2}=1$$

## 物理代考

8.1 Stern-Gerlach 实验

$$\left|c_{\uparrow}(t)\right|^{2}+\left|c_{\downarrow}(t)\right|^{2}=1$$

${ }^{1}$ 这里的状态 $|\uparrow\rangle$ 和 $|\downarrow\rangle$ 是抽象占用数中的单粒子状态

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$$\left|\psi_{\text {int }}(t)\right\rangle=c(t)|\uparrow\rangle \quad ;|c(t)|^{2}=1$$

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