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物理代写|量子力学代写Quantum mechanics代考|PHYS3001 Multiparticle Wave Functions

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物理代写|量子力学代写Quantum mechanics代考|Multiparticle Wave Functions

In a system of $N$ particles, the positions of the individual particles are independent observables that commute with one another, so a complete set (ignoring spin for now) consists of the operators $\left(\hat{\mathbf{x}}_1, \ldots, \hat{\mathbf{x}}_N\right)$, with eigenvalues $\left(\mathbf{x}_1, \ldots, \mathbf{x}_N\right)$. In this case the wave function is defined by
$$\psi\left(\mathbf{x}_1, \ldots, \mathbf{x}_N\right)=\left\langle\mathbf{x}_1, \ldots, \mathbf{x}_N \mid \psi\right\rangle$$
The wave function is defined over configuration space, that is, the space in which a single point specifies the positions of all the particles. This is the same configuration space as in classical mechanics, which is discussed in Sec. B.3. Configuration space coincides with physical space only in the case of a single particle. Similarly, one can define a multiparticle, momentum space wave function $\phi\left(\mathbf{p}_1, \ldots, \mathbf{p}_N\right)$.

If one is careful about the application of the postulates of quantum mechanics, one will see certain subtleties in the derivation of Eq. (89) in the case of identical particles. We will examine this question more carefully in Notes 29.

物理代写|量子力学代写Quantum mechanics代考|The Sign of i

The following is a remark concerning the the definition of $\hat{\mathbf{k}}$ in Eq. (30), which led to the definition of momentum in Eq. (65). We split off a factor of $i$ in Eq. (30) to make $\hat{\mathbf{k}}$ Hermitian, but the same would have been achieved if we had split off $-i$ (thereby changing the definition of $\hat{\mathbf{k}}$ by a sign). This would lead to the opposite sign in the definition of $\hat{\mathbf{p}}$, and changes in signs in many of the subsequent formulas. Would this change lead to any physical consequences or contradictions with experiment?

The answer is no, but it would change most of the familiar formulas in quantum mechanics, by replacing $i$ by $-i$. For example, a plane wave with wave vector $\mathbf{k}$ would become $e^{-i \mathbf{k} \cdot \mathbf{x}}$ instead of the usual $e^{i \mathbf{k} \cdot \mathbf{x}}$. It is a matter of convention to choose the sign of $i$ in quantum mechanics, and our choice has been made in Eqs. (30) and (65). Once this choice has been made, however, then the sign of the Pauli matrix $\sigma_y$ is determined, so that spin angular momentum has the same commutation relations as orbital angular momentum. This was a question addressed in Prob. 3.2(d).

物理代写|量子力学代写Quantum mechanics代考|Multiparticle Wave Functions

$$\psi\left(\mathbf{x}_1, \ldots, \mathbf{x}_N\right)=\left\langle\mathbf{x}_1, \ldots, \mathbf{x}_N \mid \psi\right\rangle$$

MATLAB代写

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