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We discuss a method to obtain closed-form expressions of

Physicists are quite often faced with the task of calculating

An archetypical example is the Hamiltonian _{x}_{y}_{z}

Let us recall how most textbooks of quantum mechanics proceed to demonstrate _{k} A^{k}_{i}σ_{j}_{ij}I_{ijk}σ_{k}^{2}^{n}^{2}^{n}^{+1} =

Although this standard demonstration is a relatively simple one, it seems to be tightly related to the particular properties of the operator ^{k}^{0}, ^{n}^{−1}. Thus, any infinite series, such as the one corresponding to exp ^{0}, ^{n}^{−1}. By exploiting this fact one can recover

The rest of the paper is organized as follows. First, we present Leonard's technique in a way that somewhat differs from the approach used in [

Consider the coupled system of differential equations, given by
_{1},…, _{n}^{T}^{At}^{k}e^{At}^{k}e^{At}^{At}^{n}_{n}_{−1}^{n}^{−1} + … + _{1}_{0})^{At}^{At}^{At}^{0}, ^{n}^{−1}. Let us consider the matrix
_{k}_{k}_{k}^{At}^{k}e^{At}_{t}_{=0}= ^{k}e^{At}_{t}_{=0}= ^{k}^{At}^{0},…, ^{n}^{−1}Φ(0) = ^{n}^{−1}. It is then clear that we must take the following initial conditions:
^{At}^{At}

Summarizing, the method consists in solving the _{0} + _{1}_{m}_{−1}^{m}^{−1})^{λt}_{k}

Let us return to _{k}_{k}_{k}_{k}_{k}_{k}_{k} a_{k}_{k}_{k}

Let us consider the 2 × 2 case _{±}〉. That is, _{±}〉 = ± |_{±}〉. We need no more than this to get _{+} 〉 〈_{+}| − |_{−}〉 〈_{−}| and _{+}〉 〈_{+}| + |_{−}〉 〈_{−;}|, it follows that |_{±} 〉 〈_{±}| = (_{k} exp _{k}_{k}_{k}_{±}〉 and eigenvalues ±_{±}〉 〈 _{±}|, or in terms of

Let us now see how the above method generalizes when dealing with higher-dimensional spaces. To this end, we keep dealing with rotations. The operator exp (_{i}_{i}_{i}_{j}_{ijk}X_{k}

_{i}_{0}〉 and |_{±} 〉, respectively. Similarly to the spin case, we have now
_{k}_{k}_{±}〉 〈_{±}| = (∓ ^{2}) /2, and |_{0}〉 〈_{0}| = ^{2}. Thus, we have
^{T}^{2}

The general case is now clear. Consider an operator _{k}^{0} = _{k}_{k}_{k}_{k} a_{k}_{k}_{k}_{k}_{k}^{2},…, ^{N}^{−1}. To this end, we must solve the system
_{k}_{k}^{N}^{−1}, we can express any analytic function of

The general solution can be written in terms of the inverse of the Vandermonde matrix _{j}_{j}_{j}^{k}_{k}_{+1}, with ^{−1}, the inverse of the Vandermonde matrix. This matrix inverse can be calculated as follows [_{j}_{j,k}_{j,k}^{−1}. Indeed, setting _{i}_{j}_{j}

So far, we have assumed that the eigenvalues of _{1} and λ_{2}, which are two-fold degenerate. We can group the projectors as follows:

Let us now see how the method works when applied to some well-known cases. Henceforth, we refer to the method as the Cayley–Hamilton (CH)-method, for short. Our aim is to show the simplicity of the required calculations, as compared with standard techniques.

The Foldy–Wouthuysen transformation is introduced [^{T}^{2}. Here, _{x}_{y}_{z}

The Foldy–Wouthuysen transformation is given by

We can calculate ^{2} = 1, the above matrices share the characteristic equation λ^{2} + 1 = 0. Their eigenvalues are thus ±_{1}_{,}_{2} = ±

The standard way to get this result requires developing the exponential in a power series. Thereafter, one must exploit the commutation properties of

The dynamics of several classical and quantum systems is ruled by equations that can be cast as differential equations for a three-vector _{0},
_{±}〉 and |_{0}〉 of _{3}, _{3} are, in turn, analogous to those of Pauli's _{y}_{3}) exp(_{2}) to the eigenvectors |±〉 and |0〉, thereby getting |_{±}〉 and |_{0}〉, respectively. All these calculations are easily performed using the CH-method.

Once we have |_{±}〉 and |_{0}〉, we also have the transformation matrix ^{−1}_{D}_{−}〉, |_{0}〉 and |_{+}〉. After we have carried out all calculations in the eigenbasis of _{0}. The eigenbasis of _{k}_{k}^{2} and ^{2} = −2λΩ^{3}^{2}Ω^{2})(Ω^{2}, and replacing the solution of the system _{0} in the original basis {|_{0} = _{0} and ^{2}_{0} = _{0}) – _{0}. _{0} asymptotically aligns with

The case ∂

We address now a system composed by a two-level atom and a quantized (monochromatic) electromagnetic field. Under the dipole and the rotating-wave approximations, the Hamiltonian of this system reads (in standard notation)
_{0} = _{0}^{†}^{†}_{−} + _{+}) couples the statesand |_{n}H_{n}_{n}_{n}_{0} –

A standard way [_{1} + _{2}, with _{1} = ^{†}_{z}_{2} = _{z}_{−} + _{+}). Because [_{1}, _{2}] = 0, the evolution operator can be factored as _{1}_{2} = exp(−_{1}_{2}_{2} can be obtained as well. As can be seen, this method depends on the realization that

Let us now calculate _{n} H_{n}_{n}_{m}_{n}U_{n}_{n}_{n}t_{0} ± λ). We have thus

In our case, _{n}

Replacing _{n}

This matrix is a representation in subspace Span{|^{2}^{†}^{2}/4. Proceeding similarly with the other operators that enter

As a further application, let us consider the representation of Lorentz transformations in the space of bispinors. In coordinate space, Lorentz transformations are given by
^{μν}^{00} = −^{11} = −^{22} = ^{33} = 1, ^{μν}^{μν}^{νμ}_{μ}_{μν}γ^{ν}^{μ}γ^{ν}^{μ}γ^{ν}^{μν}_{μ}γ_{ν}

For the following, it will be advantageous to define
_{i}_{i}_{5} := _{0}_{1}_{2}_{3} satisfies
_{5} (_{i}_{i}_{i}_{5}. Noting that _{i}_{i}_{i}_{i}_{i}_{k}_{k}_{k}_{i}_{j}_{ijk}σ_{k}_{i}σ_{j}_{j}σ_{i}_{ij}_{i}σ_{j}_{ijk}σ_{k}_{ij}

We can write now
_{i}_{i}^{i}^{0}^{i}^{k}ϵ_{ijk}^{ij}

Considering the isomorphism _{k}_{k}^{2} ≡ ^{2} ≡ ^{3} = ^{2}^{4} = ^{4}, ^{5} = ^{4}

As in the previous examples, also in this case the above result can be obtained more directly by noting that
_{±}〉 for the corresponding eigenvectors, _{±}〉 = _{±} |_{±}〉, we have that
_{±}〉 〈_{±}|, we get

We apply now the general decomposition exp _{n}_{n}_{n}_{n}_{±}> and eigenvalues exp (∓^{3},

The method presented in this paper—referred to as the Cayley–Hamilton method—proves advantageous for calculating closed-form expressions of analytic functions

The author declares no conflict of interest.

The author gratefully acknowledges the Research Directorate of the Pontificia Universidad Católica del Perú (DGI-PUCP) for financial support under Grant No. 2014-0064.