B.16 Chapter 36

B.16 Chapter 36

Concept links · terms present in this machine draft; source roles are unverified: vector space · complex inner product

Problem 1:

When the single-particle state space ${\mathcal{H}}_1$ is a complex vector space with Hermitian inner product, one has an infinite dimensional case of the situation of section 26.4. In this case one can write annihilation and creation operators acting on the multi-particle state spaces $S^{\ast }({\mathcal{H}}_1)$ or ${\Lambda }^{\ast }({\mathcal{H}}_1)$ in a basis independent manner as follows:

$$a^{†}(f)P^{\pm }\left(g_1\otimes g_2\otimes \cdots \otimes g_n\right)=\sqrt{n+1}{P}^{\pm }\left(f\otimes g_1\otimes g_2\otimes \cdots \otimes g_n\right)$$ $$a(f)P^{\pm }\left(g_1\otimes g_2\otimes \cdots \otimes g_n\right)=\frac{1}{\sqrt{n}}\sum _{j=1}^n(\pm 1{)}^{j+1}⟨f,g_j⟩P^{\pm }\left(g_1\otimes g_2\cdots \otimes {\hat{g}}_j\otimes \cdots \otimes g_n\right)$$

Here $f,g_j\in {\mathcal{H}}_1,P^{\pm }$ is the operation of summing over permutations used in section 9.6 that produces symmetric or antisymmetric tensor products, and $\widehat{g_j}$ means omit the $g_j$ term in the tensor product.

Show that, for f orthonormal basis elements of ${\mathcal{H}}_1$ , these annihilation and creation operators satisfy the CCR (+ case) or CAR (− case).

Problem 2:

In the fermionic case of problem 1, show that the inner product on ${\Lambda }^{\ast }({\mathcal{H}}_1)$ that, for an orthonormal basis $e_1,\cdots ,e_n$ of ${\mathcal{H}}_1$ , makes the $e_{i_1}\wedge e_{i_2}\wedge \cdots \wedge e_{i_k}$ orthonormal for $i_1<i_2<\cdots <i_k$ can be written in an basis independent way as

$$⟨f_1\wedge f_2\wedge \cdots \wedge f_k,g_1\wedge g_2\wedge \cdots \wedge g_k⟩=\det M$$

where $M$ is the k by $k$ matrix with lm entry $⟨f_l,g_m⟩$

When ${\mathcal{H}}_1$ is a space of wavefunctions (in position or momentum space), then taking $f_j$ to be some single-particle wavefunctions, and the $g_j$ to be deltafunctions in position or momentum space, this construction is known as the “Slater determinant” construction giving antisymmetric wavefunctions. For the bosonic case, a similar construction exists for symmetric tensor products, using instead of the determinant of the matrix, something called the “permanent” of the matrix.

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正文：英文 · OCR 机器稿 · 待校对

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原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

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