Subject: Time Evolution Operator

Post date: April 22, 2017

In this page we will review the basic concepts of propagation in the context of
quantum mechanics. Beginning with the Schrodinger equation:
\begin{eqnarray}
i\partial_t U(t,t_0) |\psi(0)\rangle = H(t)U(t,t_0)|\psi(t_0)\rangle, \,\,\,\, \forall |\psi(t_0)\rangle
\end{eqnarray}
where $$U$$ is the unitary time-evolution operator, $$|\psi\rangle$$ the wave function, and $$H$$ is the Hamiltonian,
we identify that
\begin{eqnarray}
i\partial_t U(t,t_0) = H(t)U(t,t_0)
\end{eqnarray}
should hold as an identity, with the initial condition of $$U(t_0,t_0)= 1$$.

In reality the system has a time-independent Hamiltonian part
and likely a perturbed component that varies in time, the two constitute $$H(t)$$ (\)=H_0 V(t)\)).

Thus,
\begin{eqnarray}
U(t+\delta,t_0) - U(t,t_0) &=& - i \delta H(t) U(t,t_0) \nonumber \\
U(t+\delta,t_0) &=& (1 - i \delta H(t)) U(t,t_0),
\end{eqnarray}

iterating the above identity with discretized time interval $$[t_0,t]$$,
\begin{eqnarray}
U(t,t_0) &=& \prod_{m=1}^{m=N} (1 - i \delta_m H(t_m)) U(t_0,t_0) \nonumber \\
U(t_0+ N\delta,t_0) &=& \prod_{m=1}^{m=N} (1 - i \delta H(m\delta)) \nonumber \\
&=& 1 -i \delta \sum_m H(m\delta)) - \delta^2 \sum_{n>m} H(n\delta) H(m\delta) \nonumber \\
&+& \dots
\end{eqnarray}

Since $$\sum_{n>m} + \sum_{n<m} \rightarrow \sum_{n,m}$$ in the limit of $$\delta \rightarrow 0$$ we have

\begin{eqnarray}
U(t_0 + N\delta,t_0) &=& 1 -i \delta \sum_m H(m\delta)
- \frac{1}{2} \delta^2 \sum_{n,m} \hat{T}[H(n\delta) H(m\delta)] \nonumber \\
&+& \dots \nonumber \\
&\rightarrow& 1 -i \int_{t_0}^t ds H(s) \nonumber \\
&+& \frac{(-i)^2}{2!} \int_{t_0}^t \int_{t_0}^t ds_1 ds_2 \hat{T} [H(s_1) H(s_2)] \nonumber \\
&+& \dots \nonumber \\
U(t ,t_0) &:=& \hat{T} e^{-i\int_{t_0}^t H(s) ds}
\end{eqnarray}
where $$\hat{T}$$ stands for the time ordering operator and means the operators must be written in order of their times with earlier time on the right. The last line is a symbolic representation. The exponential $$e^{i(\dots)}$$ with $$\dots$$ a Hermitian operator (that is "real" eigenvalues),
emphasizes that the operator $$U$$ is a unitary operator (preserving the norm).

However, as one can visually see, the truncated expansion does not seem to be
of norm one. This result in the norm of the state
$$U(t,t_0)|\psi\rangle$$ to vary from one. Thus the expansion must be accompanied by a denominator $$\frac{U|\psi\rangle}{\langle \psi | U^\dagger U |\psi\rangle}$$.

In the adiabatic regime (non-degenerate), Gell-Mann and Low proved that the state
$$\frac{U|\psi\rangle}{\langle \psi | U^\dagger U |\psi\rangle}$$is an eigen state of the $$H(t)$$ as the
time-varying part of the Hamiltonian is adiabatically
turned on . This is in particular important in the study of the ground state
of interacting many-body systems where the $$|\psi\rangle$$ is the ground state of the non-interacting system and is
adiabatically perturbed to an interacting regime.

Post date: May 14, 2017

Here is a quick demonstration of Bias-Variance Trade-Off.
I am going to use some actual data. The goal is to fit a polynomial function to the data points.
I call the target data as $$y$$. The index to the data $$x$$ is normalized so they run from $$[0,1]$$.

We are going to first pick a subset, the training set, and use them to find the correct
parameters of the model. The model is a polynomial function, $$f(x)=a_0 + \dots a_N x^N$$.
The maximum power $$N$$ that needs to be used in the model is unknown. So, we are going to
use the Bias-Variance Trade-Off concept to find what is the best power to be used.

Here I use half of the data points to train the model. The data looks something like this:

Using this set I find the parametrs $A$ such that $$X(x) \,\, A= Y_t.$$ here $$X$$ is a matrix width rows that goes like $$\left[x^0 \dots \, x^{N} \right]$$ points according to the maximum power of the polynomial. For a max power of $$4$$ the plot looks like

repeating the process for a set of models created as explained above we can find that the R-MSE (Root of the Mean Square Error) looks something like this: