Re: Ploting a transformation of a set
- To: mathgroup at smc.vnet.net
- Subject: [mg123497] Re: Ploting a transformation of a set
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Fri, 9 Dec 2011 05:55:18 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <4EDCB5F0.813B.006A.0@newcastle.edu.au>
- Reply-to: drmajorbob at yahoo.com
The short (or not so short) answer is:
Eigenvalues@{{-1/2*S22*\[Chi]^2 + I*a21*\[Chi],
I/(2*\[HBar])*\[CapitalDelta], -I/(2*\[HBar])*\[CapitalDelta],
0}, {I/(2*\[HBar])*\[CapitalDelta],
2*S12/\[HBar] - S22*X^2/2 - 2*S11/\[HBar] - I*\[Epsilon]/\[HBar] -
2*I*\[Lambda]/\[HBar],
0, -I/(2*\[HBar])*\[CapitalDelta]}, \
{-I/(2*\[HBar])*\[CapitalDelta],
0, -2*S12/\[HBar] - S22*X^2/2 - 2*S11/\[HBar] +
I*\[Epsilon]/\[HBar] + 2*I*\[Lambda]/\[HBar],
I/(2*\[HBar])*\[CapitalDelta]}, {0, -I/(2*\[HBar])*\[CapitalDelta],
I/(2*\[HBar])*\[CapitalDelta], -1/2*S22*\[Chi]^2 + I*a21*\[Chi]}}
{-(1/2) \[Chi] (-2 I a21 + S22 \[Chi]),
Root[16 S11 \[CapitalDelta]^2 +
4 S22 X^2 \[CapitalDelta]^2 \[HBar] -
32 I a21 S11^2 \[Chi] \[HBar] + 32 I a21 S12^2 \[Chi] \[HBar] +
32 a21 S12 \[Epsilon] \[Chi] \[HBar] -
8 I a21 \[Epsilon]^2 \[Chi] \[HBar] +
64 a21 S12 \[Lambda] \[Chi] \[HBar] -
32 I a21 \[Epsilon] \[Lambda] \[Chi] \[HBar] -
32 I a21 \[Lambda]^2 \[Chi] \[HBar] +
16 S11^2 S22 \[Chi]^2 \[HBar] - 16 S12^2 S22 \[Chi]^2 \[HBar] +
16 I S12 S22 \[Epsilon] \[Chi]^2 \[HBar] +
4 S22 \[Epsilon]^2 \[Chi]^2 \[HBar] +
32 I S12 S22 \[Lambda] \[Chi]^2 \[HBar] +
16 S22 \[Epsilon] \[Lambda] \[Chi]^2 \[HBar] +
16 S22 \[Lambda]^2 \[Chi]^2 \[HBar] -
16 I a21 S11 S22 X^2 \[Chi] \[HBar]^2 +
8 S11 S22^2 X^2 \[Chi]^2 \[HBar]^2 -
2 I a21 S22^2 X^4 \[Chi] \[HBar]^3 +
S22^3 X^4 \[Chi]^2 \[HBar]^3 + 32 S11^2 \[HBar] #1 -
32 S12^2 \[HBar] #1 + 8 \[CapitalDelta]^2 \[HBar] #1 +
32 I S12 \[Epsilon] \[HBar] #1 + 8 \[Epsilon]^2 \[HBar] #1 +
64 I S12 \[Lambda] \[HBar] #1 +
32 \[Epsilon] \[Lambda] \[HBar] #1 + 32 \[Lambda]^2 \[HBar] #1 +
16 S11 S22 X^2 \[HBar]^2 #1 - 32 I a21 S11 \[Chi] \[HBar]^2 #1 +
16 S11 S22 \[Chi]^2 \[HBar]^2 #1 + 2 S22^2 X^4 \[HBar]^3 #1 -
8 I a21 S22 X^2 \[Chi] \[HBar]^3 #1 +
4 S22^2 X^2 \[Chi]^2 \[HBar]^3 #1 + 32 S11 \[HBar]^2 #1^2 +
8 S22 X^2 \[HBar]^3 #1^2 - 8 I a21 \[Chi] \[HBar]^3 #1^2 +
4 S22 \[Chi]^2 \[HBar]^3 #1^2 + 8 \[HBar]^3 #1^3 &, 1],
Root[16 S11 \[CapitalDelta]^2 +
4 S22 X^2 \[CapitalDelta]^2 \[HBar] -
32 I a21 S11^2 \[Chi] \[HBar] + 32 I a21 S12^2 \[Chi] \[HBar] +
32 a21 S12 \[Epsilon] \[Chi] \[HBar] -
8 I a21 \[Epsilon]^2 \[Chi] \[HBar] +
64 a21 S12 \[Lambda] \[Chi] \[HBar] -
32 I a21 \[Epsilon] \[Lambda] \[Chi] \[HBar] -
32 I a21 \[Lambda]^2 \[Chi] \[HBar] +
16 S11^2 S22 \[Chi]^2 \[HBar] - 16 S12^2 S22 \[Chi]^2 \[HBar] +
16 I S12 S22 \[Epsilon] \[Chi]^2 \[HBar] +
4 S22 \[Epsilon]^2 \[Chi]^2 \[HBar] +
32 I S12 S22 \[Lambda] \[Chi]^2 \[HBar] +
16 S22 \[Epsilon] \[Lambda] \[Chi]^2 \[HBar] +
16 S22 \[Lambda]^2 \[Chi]^2 \[HBar] -
16 I a21 S11 S22 X^2 \[Chi] \[HBar]^2 +
8 S11 S22^2 X^2 \[Chi]^2 \[HBar]^2 -
2 I a21 S22^2 X^4 \[Chi] \[HBar]^3 +
S22^3 X^4 \[Chi]^2 \[HBar]^3 + 32 S11^2 \[HBar] #1 -
32 S12^2 \[HBar] #1 + 8 \[CapitalDelta]^2 \[HBar] #1 +
32 I S12 \[Epsilon] \[HBar] #1 + 8 \[Epsilon]^2 \[HBar] #1 +
64 I S12 \[Lambda] \[HBar] #1 +
32 \[Epsilon] \[Lambda] \[HBar] #1 + 32 \[Lambda]^2 \[HBar] #1 +
16 S11 S22 X^2 \[HBar]^2 #1 - 32 I a21 S11 \[Chi] \[HBar]^2 #1 +
16 S11 S22 \[Chi]^2 \[HBar]^2 #1 + 2 S22^2 X^4 \[HBar]^3 #1 -
8 I a21 S22 X^2 \[Chi] \[HBar]^3 #1 +
4 S22^2 X^2 \[Chi]^2 \[HBar]^3 #1 + 32 S11 \[HBar]^2 #1^2 +
8 S22 X^2 \[HBar]^3 #1^2 - 8 I a21 \[Chi] \[HBar]^3 #1^2 +
4 S22 \[Chi]^2 \[HBar]^3 #1^2 + 8 \[HBar]^3 #1^3 &, 2],
Root[16 S11 \[CapitalDelta]^2 +
4 S22 X^2 \[CapitalDelta]^2 \[HBar] -
32 I a21 S11^2 \[Chi] \[HBar] + 32 I a21 S12^2 \[Chi] \[HBar] +
32 a21 S12 \[Epsilon] \[Chi] \[HBar] -
8 I a21 \[Epsilon]^2 \[Chi] \[HBar] +
64 a21 S12 \[Lambda] \[Chi] \[HBar] -
32 I a21 \[Epsilon] \[Lambda] \[Chi] \[HBar] -
32 I a21 \[Lambda]^2 \[Chi] \[HBar] +
16 S11^2 S22 \[Chi]^2 \[HBar] - 16 S12^2 S22 \[Chi]^2 \[HBar] +
16 I S12 S22 \[Epsilon] \[Chi]^2 \[HBar] +
4 S22 \[Epsilon]^2 \[Chi]^2 \[HBar] +
32 I S12 S22 \[Lambda] \[Chi]^2 \[HBar] +
16 S22 \[Epsilon] \[Lambda] \[Chi]^2 \[HBar] +
16 S22 \[Lambda]^2 \[Chi]^2 \[HBar] -
16 I a21 S11 S22 X^2 \[Chi] \[HBar]^2 +
8 S11 S22^2 X^2 \[Chi]^2 \[HBar]^2 -
2 I a21 S22^2 X^4 \[Chi] \[HBar]^3 +
S22^3 X^4 \[Chi]^2 \[HBar]^3 + 32 S11^2 \[HBar] #1 -
32 S12^2 \[HBar] #1 + 8 \[CapitalDelta]^2 \[HBar] #1 +
32 I S12 \[Epsilon] \[HBar] #1 + 8 \[Epsilon]^2 \[HBar] #1 +
64 I S12 \[Lambda] \[HBar] #1 +
32 \[Epsilon] \[Lambda] \[HBar] #1 + 32 \[Lambda]^2 \[HBar] #1 +
16 S11 S22 X^2 \[HBar]^2 #1 - 32 I a21 S11 \[Chi] \[HBar]^2 #1 +
16 S11 S22 \[Chi]^2 \[HBar]^2 #1 + 2 S22^2 X^4 \[HBar]^3 #1 -
8 I a21 S22 X^2 \[Chi] \[HBar]^3 #1 +
4 S22^2 X^2 \[Chi]^2 \[HBar]^3 #1 + 32 S11 \[HBar]^2 #1^2 +
8 S22 X^2 \[HBar]^3 #1^2 - 8 I a21 \[Chi] \[HBar]^3 #1^2 +
4 S22 \[Chi]^2 \[HBar]^3 #1^2 + 8 \[HBar]^3 #1^3 &, 3]}
Those are the roots of
d = Det@{{-1/2*S22*\[Chi]^2 + I*a21*\[Chi],
I/(2*\[HBar])*\[CapitalDelta], -I/(2*\[HBar])*\[CapitalDelta],
0}, {I/(2*\[HBar])*\[CapitalDelta],
2*S12/\[HBar] - S22*X^2/2 - 2*S11/\[HBar] -
I*\[Epsilon]/\[HBar] - 2*I*\[Lambda]/\[HBar],
0, -I/(2*\[HBar])*\[CapitalDelta]}, {-I/(2*\[HBar])*\
\[CapitalDelta],
0, -2*S12/\[HBar] - S22*X^2/2 - 2*S11/\[HBar] +
I*\[Epsilon]/\[HBar] + 2*I*\[Lambda]/\[HBar],
I/(2*\[HBar])*\[CapitalDelta]}, {0, \
-I/(2*\[HBar])*\[CapitalDelta],
I/(2*\[HBar])*\[CapitalDelta], -1/2*S22*\[Chi]^2 +
I*a21*\[Chi]}} // Factor
-(1/(16 \[HBar]^3))\[Chi] (2 a21 +
I S22 \[Chi]) (16 I S11 \[CapitalDelta]^2 +
4 I S22 X^2 \[CapitalDelta]^2 \[HBar] +
32 a21 S11^2 \[Chi] \[HBar] - 32 a21 S12^2 \[Chi] \[HBar] +
32 I a21 S12 \[Epsilon] \[Chi] \[HBar] +
8 a21 \[Epsilon]^2 \[Chi] \[HBar] +
64 I a21 S12 \[Lambda] \[Chi] \[HBar] +
32 a21 \[Epsilon] \[Lambda] \[Chi] \[HBar] +
32 a21 \[Lambda]^2 \[Chi] \[HBar] +
16 I S11^2 S22 \[Chi]^2 \[HBar] -
16 I S12^2 S22 \[Chi]^2 \[HBar] -
16 S12 S22 \[Epsilon] \[Chi]^2 \[HBar] +
4 I S22 \[Epsilon]^2 \[Chi]^2 \[HBar] -
32 S12 S22 \[Lambda] \[Chi]^2 \[HBar] +
16 I S22 \[Epsilon] \[Lambda] \[Chi]^2 \[HBar] +
16 I S22 \[Lambda]^2 \[Chi]^2 \[HBar] +
16 a21 S11 S22 X^2 \[Chi] \[HBar]^2 +
8 I S11 S22^2 X^2 \[Chi]^2 \[HBar]^2 +
2 a21 S22^2 X^4 \[Chi] \[HBar]^3 + I S22^3 X^4 \[Chi]^2 \[HBar]^3)
The SIMPLE factors give these eigenvalues:
Solve[Take[d, 4] == 0]
{{a21 -> -(1/2) I S22 \[Chi]}, {\[Chi] -> 0}}
Leaving those out of it and concentrating on REAL variables, other
eigenvalues might come from Solve, as follows.
\[HBar]^3 doesn't factor out of the 5th factor in d:
ComplexExpand@Through[{Re, Im}@d[[5]]] // Factor
{2 \[Chi] \[HBar] (16 a21 S11^2 - 16 a21 S12^2 + 4 a21 \[Epsilon]^2 +
16 a21 \[Epsilon] \[Lambda] + 16 a21 \[Lambda]^2 -
8 S12 S22 \[Epsilon] \[Chi] - 16 S12 S22 \[Lambda] \[Chi] +
8 a21 S11 S22 X^2 \[HBar] + a21 S22^2 X^4 \[HBar]^2),
16 S11 \[CapitalDelta]^2 + 4 S22 X^2 \[CapitalDelta]^2 \[HBar] +
32 a21 S12 \[Epsilon] \[Chi] \[HBar] +
64 a21 S12 \[Lambda] \[Chi] \[HBar] +
16 S11^2 S22 \[Chi]^2 \[HBar] - 16 S12^2 S22 \[Chi]^2 \[HBar] +
4 S22 \[Epsilon]^2 \[Chi]^2 \[HBar] +
16 S22 \[Epsilon] \[Lambda] \[Chi]^2 \[HBar] +
16 S22 \[Lambda]^2 \[Chi]^2 \[HBar] +
8 S11 S22^2 X^2 \[Chi]^2 \[HBar]^2 + S22^3 X^4 \[Chi]^2 \[HBar]^3}
Hence, we have to assume \[HBar] is nonzero. We can also assume \[Chi] is
nonzero, since we already have a root corresponding to \[Chi] = 0.
vars = Variables@d
{\[Chi], a21, S22, \[HBar], S11, \[CapitalDelta], X, S12, \[Epsilon], \
\[Lambda]}
eqns = Flatten@{ComplexExpand@Through[{Re, Im}@d[[5]]] == 0 //
Thread, \[HBar] != 0, \[Chi] != 0};
Solve[eqns]
(* long output *)
You can change that to solve for your preferred variables in terms of
others (not the choices Solve makes for us).
Bobby
On Thu, 08 Dec 2011 04:27:52 -0600, ç?? ç«¥ <z.tong1017 at gmail.com> wrote:
> hello,can u help me solve a matrix problem~
> A1 = {{-1/2*S22*\[Chi]^2 + I*a21*\[Chi],
> I/(2*\[HBar])*\[CapitalDelta], -I/(2*\[HBar])*\[CapitalDelta],
> 0}, {I/(2*\[HBar])*\[CapitalDelta],
> 2*S12/\[HBar] - S22*X^2/2 - 2*S11/\[HBar] -
> I*\[Epsilon]/\[HBar] - 2*I*\[Lambda]/\[HBar],
> 0, -I/(2*\[HBar])*\[CapitalDelta]}, {-I/(2*\[HBar])*\
> \[CapitalDelta],
> 0, -2*S12/\[HBar] - S22*X^2/2 - 2*S11/\[HBar] +
> I*\[Epsilon]/\[HBar] + 2*I*\[Lambda]/\[HBar],
> I/(2*\[HBar])*\[CapitalDelta]}, {0, \
> -I/(2*\[HBar])*\[CapitalDelta],
> I/(2*\[HBar])*\[CapitalDelta], -1/2*S22*\[Chi]^2 +
> I*a21*\[Chi]}} // MatrixForm
> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
> Solve the eigenvalues of the matrix A1,thx!~
>
--
DrMajorBob at yahoo.com