- #1
umut_caglar
- 7
- 0
Hi guys I have a problem in finding my error in a calculation, I will be glad if you help me to find the error that I am doing
ok the problem is basically about matrix exponentials, here we go:
A, B, U, P are matrices
n is a natural number
t and T are rational numbers and T=n*t
now in general ## e^{t(A+B)}≠e^{tA}*e^{tB} ##
but can be represented by using The Zassenhaus formula
## e^{t(A+B)}=e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots ##
one can find the details of the formula from http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
now I begin by writing the formula
## e^{t(A+B)}=e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots ##
and then I take the 'n'th power of both sides
## \left(e^{t(A+B)}\right)^n=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##if I define U=A+B I will get
## \left(e^{t U}\right)^n=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##
by using the equality of (e^A)^n=e^[aN] i will obtain
## e^{t*n*U}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##
by using the definitions T=nt and U=A+B i will get
## e^{T*(A+B)}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##
for the right side I define the P matrix as P=AB-BA
## e^{T*(A+B)}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*P}*\ldots\right)^n ##
and by using the equality ## \left(e^A*e^B\right)^n=e^{(nA)}*e^{(nB)} ## recursively, I obtain
## e^{T*(A+B)}=e^{t*n*A}*e^{t*n*B}*e^{(t^2/2)*n*P}*\ldots ##
Again by using the definitions of T=tn and P=AB-BA I obtain
## e^{T*(A+B)}=e^{T*A}*e^{T*B}*e^{T*(t/2)*(AB-BA)}*\ldots ##
but, if expand the left side of the equation by using Zassenhaus formula, I end up with
## e^{TA}*e^{TB}*e^{(T^2/2)(AB-BA)}*...≠e^{TA}*e^{TB}*e^{(T*t/2)(AB-BA)}*\ldots ##which is not clearly equal to the right side; so where is my mistake.*****************************
As a side note; If you also show the correct version of the calculation I will be glad;
For example; I expect my error is asuuming the equality of ## \left(e^A*e^B\right)^n=e^{(nA)}*e^{(nB)} ##
if this is the mistake, could you show the correct relation?
Thanks for the help
ok the problem is basically about matrix exponentials, here we go:
A, B, U, P are matrices
n is a natural number
t and T are rational numbers and T=n*t
now in general ## e^{t(A+B)}≠e^{tA}*e^{tB} ##
but can be represented by using The Zassenhaus formula
## e^{t(A+B)}=e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots ##
one can find the details of the formula from http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
now I begin by writing the formula
## e^{t(A+B)}=e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots ##
and then I take the 'n'th power of both sides
## \left(e^{t(A+B)}\right)^n=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##if I define U=A+B I will get
## \left(e^{t U}\right)^n=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##
by using the equality of (e^A)^n=e^[aN] i will obtain
## e^{t*n*U}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##
by using the definitions T=nt and U=A+B i will get
## e^{T*(A+B)}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##
for the right side I define the P matrix as P=AB-BA
## e^{T*(A+B)}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*P}*\ldots\right)^n ##
and by using the equality ## \left(e^A*e^B\right)^n=e^{(nA)}*e^{(nB)} ## recursively, I obtain
## e^{T*(A+B)}=e^{t*n*A}*e^{t*n*B}*e^{(t^2/2)*n*P}*\ldots ##
Again by using the definitions of T=tn and P=AB-BA I obtain
## e^{T*(A+B)}=e^{T*A}*e^{T*B}*e^{T*(t/2)*(AB-BA)}*\ldots ##
but, if expand the left side of the equation by using Zassenhaus formula, I end up with
## e^{TA}*e^{TB}*e^{(T^2/2)(AB-BA)}*...≠e^{TA}*e^{TB}*e^{(T*t/2)(AB-BA)}*\ldots ##which is not clearly equal to the right side; so where is my mistake.*****************************
As a side note; If you also show the correct version of the calculation I will be glad;
For example; I expect my error is asuuming the equality of ## \left(e^A*e^B\right)^n=e^{(nA)}*e^{(nB)} ##
if this is the mistake, could you show the correct relation?
Thanks for the help
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