# How should I solve the Schrödinger equation by diagonalization in QFT?

My professor recently send me off to solve the Anderson Impurity Model (AIM), which is expressed in terms of creation and annihilation operators: $$\widehat{H} = \sum_\sigma \varepsilon_d d_\sigma^\dagger d_\sigma + U d_\uparrow^\dagger d_\uparrow d_\downarrow^\dagger d_\downarrow + \sum_{k,\sigma} \varepsilon_k c_{k \sigma}^\dagger c_{k\sigma} + \sum_{k,\sigma} V_k(d_\sigma^\dagger c_{k\sigma} + c_{k \sigma}^\dagger d_\sigma).$$ The on-site energy of the impurity is $$\varepsilon_d$$ and the interaction term which penalizes double occupancy is $$U$$. The bath dispersion is $$\varepsilon_k$$ and the hybridization between the bath and the impurity is $$V_k$$ , where $$k$$ is a wave number.

I want to solve it by diagonalization, because it is a standard way to solve the Schrödinger equation in low energy physics. Diagonalization entails finding the zeros of the characteristic polynomial of the standard matrix of the hamiltonian relative to a finite basis for the hilbert space on which $$\widehat H$$ acts. The standard matrix of $$\widehat H$$ has matrix elements $$H_{ij} = \langle \Phi_i, \widehat H \Phi_j \rangle,$$ where $$\Phi_1,\cdots,\Phi_N$$ denotes an orthonormal finite basis for the space on which $$\widehat H$$ is defined.

My question: My textbook does not tell me how the ladder operators from AIM act on an arbitrary function. It seems it is assumed knowledge. However, in the appendix, it says how the ladders operator acts on the vacuum.

Could you say how the ladder operators from the AIM theory act on a regular function, (and not the vacuum)? That way I can find the standard matrix of the hamiltonian, and diagonalize it.

• The annihilation and creation operators commute with any numerical function such as $\epsilon_k$ or $V_k$. Jan 1, 2021 at 20:35
• Perhaps I am misunderstanding you, but can't an arbitrary function be written as a (possibly infinite) sum of vectors made by products of creation operators acting on the vacuum? Jan 2, 2021 at 0:44
• No idea. I suspect this is part of the answer. Basically I want to diagonalize this hamiltonian, and I know how to do it for operators defined on $L^2(\mathbb{R}^d)$. Jan 2, 2021 at 0:55
• There are various papers. Jan 2, 2021 at 1:03

The crucial thing to understand is that the ladder operators $$c_{k\sigma}$$ do not act on the Hilbert space $$L^2(\mathbb{R}^n)$$, but rather on the Fock space, whose basis is given by the occupation numbers: $$\left|v\right\rangle=\bigotimes_{k\sigma}\left|n_{k,\sigma}\right\rangle$$ That is, we specify an occupation number (0 or 1 for Fermions) for each mode of the system. The ladder operators are used to move us between the occupation numbers: $$\left|v\right\rangle=\prod_{k\sigma}a^\dagger_{k\sigma}\left|\Omega\right\rangle$$ and act as $$a_{k\sigma}\left|n_{k\sigma}\right\rangle=\cases{\left|0_{k\sigma}\right\rangle & n=1 \\0 & n=0}$$ $$a^\dagger_{k\sigma}\left|n_{k\sigma}\right\rangle=\cases{0 & n=1 \\\left|1_{k\sigma}\right\rangle & n=0}$$ Second quantized operators are described in basically any field theory textbook. You can consult Altland & Simons as a recommended first read.