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Question

In the following expression can $\epsilon$ be a matrix?

$$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + \epsilon^2 E_2 |m_2\rangle + \dots) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) $$

Background

So in quantum mechanics we generally have a solution $|m\rangle$ to a Hamiltonian:

$$ H | m\rangle = E |m\rangle $$

Now using perturbation theory:

$$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + \epsilon^2 E_2 |m_2\rangle + \dots) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) $$

I was curious and substituted $\epsilon$ as a matrix:

$$ \epsilon = \left( \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right) $$

where $\epsilon$ now, is the nilpotent matrix, we get:

$$ \left( \begin{array}{cc} H | m \rangle & 0 \\ H_1 |m_1 \rangle + H | m\rangle & H |m_1 \rangle \end{array} \right) = \left( \begin{array}{cc} E | m \rangle & 0 \\ E_1 |m_1 \rangle + E | m\rangle & E |m_1 \rangle \end{array} \right)$$

Which is what we'd expect if we compared powers of $\epsilon$'s. All this made me wonder if $\epsilon$ could be a matrix? Say something like $| m_k\rangle \langle m_k |$ ? Say we chose $\epsilon \to \hat I \epsilon$

then there exists a radius of convergence. What is the radius of convergence in a general case of any matrix?

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    $\begingroup$ The idea of making a quantity arbitrarily small is not exclusive to a number (you need to ask yourself, essentially what does it mean to be small?). In general, if you have a normed space the norm induces a metric on it, which gives you a notion of "distance" among its elements, so it makes sense to ask for this distance to decrease or be arbitrarily small. The space of matrices is just a particular case of this general setting, see: en.wikipedia.org/wiki/Matrix_norm $\endgroup$ – hjhjhj57 Jan 26 '16 at 22:40
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I would say there's nothing preventing you from using a matrix as perturbation of a matrix equation as long as you let the limit of the norm converge to $0$.

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$\epsilon$ is not generally used as a matrix. When matrices are needed, as you will notice from your example, epsilon is usually set beside the matrices of concern as a scalar which indicates that one is infinitesimally close to the origin (or zero) in whichever space you seek to be perturbing.

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Yes and No. When I say yes I mean it is possible in several ways. But it does not make sense.

As usual for a physicist you did not specify what the space is in which your states (kets) live and thus not what the $H$ and $H_1$ are. But of course they are meant to be operators which you can consider to be generalizations of matrices to infinite dimensions. So when you assume $\epsilon$ to be a matrix you could as well absorb this into $H_1$.

Further you put $\epsilon$ to be a constant matrix. Then you can just leave the $\epsilon$ away. The epsilon is there to control the perturbation $H_1$. If you set $\epsilon$ to be a constant you directly solve the problem.

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