Many people may think that an irrational such as 2^1/2 is mathematical, not physical, and has no direct connection to quantum mechanics (QM).
On the other hand, we guess that's a great question even though no one really knows the exact answer.
We offer the following:
For the interpretation of probabilities in QM to make sense, the wave function Ψ must satisfy certain conditions.
An extremely important and yet rarely mentioned condition is,
Ψ squared = Ψ* squared=Ψ.Ψ* must always be positive and real.
This is the required answer.
Matrix transition chains B (solving the heat diffusion/conduction equation as a function of time) suggests finding an adequate alternative complex transition matrix to solve the Schrödinger equation as a function of time.
what is quite striking is that 2^1/2 should appear explicitly and be expressed numerically as 1.142... in order to construct the required complex transition matrix.
The answer to this subtle question is not complicated but quite long.
The basic assumption here is that the transition matrix (Q) representation of the solution to Schrödinger's equation exists:
E=Q+Q^2+ . . +Q^N
and should be subject to the statistics of nature, just like the transition matrix B itself
and it is expected that,
Q=B^1/2
This can be best explained by numerical examples.
A simple
example is the time-dependent solution of a quantum particle in a one-dimensional 5-node infinite potential well.
The generating transition matrix B is given by,
0 1/2 0 0 0
1/2 0 1/2 0 0
0 1/2 0 1/2 0
0 0 1/2 0 1/2
0 0 0 1/2 0
And the complex quantum transition matrix Q = B^1/2 is,
((1+i)*2^0.5*3^(1/4)+(3+3i)*2^0.5)/24 ((1-i)*2^0.5*27^(1/4)+(3-3i)*2^0.5)/24 (1+i)*2^0.5*3^(1/4)/12 ((1-i)*2^0.5*27^(1/4)-(3- 3i)*2^0.5)/24 ((1+i)*2^0.5*3^(1/4)-(3+3i)*2^0.5)/24
((1-i)*2^0.5*27^(1/4)+(3-3i)*2^0.5)/24 ((1+i)*2^0.5*3^(1/4)+(1+i)*2^0.5)/8 (1-i)*2^0.5*27^(1/4)/12 ((1+i)*2^0.5*3^(1/4)-(1+i) *2^0.5)/8 ((1-i)*2^0.5*27^(1/4)-(3-3i)*2^0.5)/24
(1+i)*2^0.5*3^(1/4)/12 (1-i)*2^0.5*27^(1/4)/12 (1+i)*2^0.5*3^(1/4)/6 (1-i)*2^0.5*27^(1/4)/12 (1+i)*2^0.5*3^(1/4)/12
((1-i)*2^0.5*27^(1/4)-(3-3i)*2^0.5)/24 ((1+i)*2^0.5*3^(1/4)-(1+i)*2^0.5)/8 (1-i)*2^0.5*27^(1/4)/12 ((1+i)*2^0.5*3^(1/4)+(1+i) *2^0.5)/8 ((1-i)*2^0.5*27^(1/4)+(3-3i)*2^0.5)/24
((1+i)*2^0.5*3^(1/4)-(3+3i)*2^0.5)/24 ((1-i)*2^0.5*27^(1/4)-(3-3i)*2^0.5)/24 (1+i)*2^0.5*3^(1/4)/12 ((1-i)*2^0.5*27^(1/4)+(3- 3i)*2^0.5)/24 ((1+i)*2^0.5*3^(1/4)+(3+3i)*2^0.5)/24
In addition to its conjugate Q*.
Note that the state,
Ψ squared = Ψ* squared=Ψ.Ψ* is positive and real and that 2^1/2 appears explicitly as a guarantee of this condition.
To be continued.
Dear Ismail Abbas I'm sure you know, QM is a theory up to this date, which it means, it never been proven as real. On the other hand, a flat one dimension static number cannot describe three dimension nature that it is changing constantly through temperature and pressure. Reading my article might help Regards.
Humble question: why asking a question on a problem while having already a solution?
II-Answer II continued.
We recall again the real matrix transition chains B (solving the heat diffusion/conduction equation as a function of time) suggests finding an adequate alternative complex transition matrix to solve the Schrödinger equation as a function of time.
In short, we offer the following:
i- There is a complex transition matrix Q for the solution of Schrödinger's equation.
In other words, one can solve the Schrödinger equation without needing neither the Schrödinger equation itself, nor the method of separation of the variables x,y,z,t nor the finite difference method (FDM) of Newton.
ii-The proposed statistical complex transition matrix Q is given by,
Q=√ (B).
If we apply the transition matrix presented in the answer above to the 1D infinite potential well of unit V, the numerical result of the eigenvalues of the energy (which corresponds to the solution of the classical Schrödinger equation) will be:
0.0
9.321
15.483
17.641
15.483
9.321
0.0
By dividing by E=17.641 (normalization condition)
we arrive at,
1- node/energy level 1 E=0=sin(0)
2-energy level 2 E is almost equal to 0.5 =sin(Pie/6)
3- node/energy level 3=0.878 is almost equal to sqrt(3)/2=sin(Pie/3)
And
4-17.641./17.641=1.0 .=sin(90) or =sin(Pie/2)
Complies with the classic SE solution for particles in the 1D potential well.
Now:
If we move to the case of a quantum particle in a 2D infinite potential well with 9 equidistant nodes, then,
The real transition matrix B would be,
0 1/4 0 1/4 0 0 0 0 0
1/4 0 1/4 0 1/4 0 0 0 0
0 1/4 0 0 0 1/4 0 0 0
1/4 0 0 0 1/4 0 1/4 0 0
0 1/4 0 1/4 0 1/4 0 1/4 0
0 0 1/4 0 1/4 0 0 0 1/4
0 0 0 1/4 0 0 0 1/4 0
0 0 0 0 1/4 0 1/4 0 1/4
0 0 0 0 0 1/4 0 1/4 0
and its corresponding complex transition matrix Q is given by,
Q=√ (B)=
((4+4i)*2^(1/4)+(1+i)*8^(1/4))/32 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/32 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/32 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 ((-4-4i)*2^(1/4)+(1+i)*8^(1/4))/32
((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 ((2+2i)*2^(1/4)+(1+i)*8^(1/4))/16 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 (1-i)*2^(1/4)/8 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 ((-2-2i)*2^(1/4)+(1+i)*8^(1/4))/16 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16
(1+i)*8^(1/4)/32 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 ((4+4i)*2^(1/4)+(1+i)*8^(1/4))/32 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 ((-4-4i)*2^(1/4)+(1+i)*8^(1/4))/32 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/32
((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 ((2+2i)*2^(1/4)+(1+i)*8^(1/4))/16 (1-i)*2^(1/4)/8 ((-2-2i)*2^(1/4)+(1+i)*8^(1/4))/16 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16
(1+i)*8^(1/4)/16 (1-i)*2^(1/4)/8 (1+i)*8^(1/4)/16 (1-i)*2^(1/4)/8 (1+i)*8^(1/4)/8 (1-i)*2^(1/4)/8 (1+i)*8^(1/4)/16 (1-i)*2^(1/4)/8 (1+i)*8^(1/4)/16
((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 ((-2-2i)*2^(1/4)+(1+i)*8^(1/4))/16 (1-i)*2^(1/4)/8 ((2+2i)*2^(1/4)+(1+i)*8^(1/4))/16 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16
(1+i)*8^(1/4)/32 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 ((-4-4i)*2^(1/4)+(1+i)*8^(1/4))/32 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 ((4+4i)*2^(1/4)+(1+i)*8^(1/4))/32 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/32
((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 ((-2-2i)*2^(1/4)+(1+i)*8^(1/4))/16 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 (1-i)*2^(1/4)/8 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 ((2+2i)*2^(1/4)+(1+i)*8^(1/4))/16 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16
((-4-4i)*2^(1/4)+(1+i)*8^(1/4))/32 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/32 ((1-i)*2^(1/4)-(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/16 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 (1+i)*8^(1/4)/32 ((1-i)*2^(1/4)+(1-i)*8^(1/4))/16 ((4+4i)*2^(1/4)+(1+i)*8^(1/4))/32
The time-independent steady-state transfer matrix (we call it W) will be given by,
W=Q+Q^2+Q^3+. . . . . Q^N . . (1)
for N sufficiently large.
Note that for N large enough, the expression 1 reduces to,
W=1/(I-Q)
It is simple to show that the 9 eigenvalues are given by,
0.613 0.866 0.613
0.866 1.225 0.866
0.613 0.866 0.613 . . . . . Distribution 1
Note that distribution 1 conforms precisely to the classical solution of SE in the 2D infinite potential well (two sin functions).
Here the subtle question arises:
So far we have dealt with the solution of the transition matrix Q to the Schrodinger equation in Cartesian coordinates but
what about the solution of the transition matrix Q with Schrodinger's equation for a particle in the central field (spherical coordinates) and the resulting quantum numbers n, l . . etc ?
This will be the subject of the next answer.
To be continued.
III-Answer iii continued
Below, we present the first published relationship between classical physics and quantum physics.This connection or relation merges from the solution of the heat conduction equation as a function of time via the real statistical transition matrix B with the solution of the equation of the Shrödinger equation as a function of time via the matrix of complex statistical transition Q.In short, this connection or link is rigidly expressed by:Q=√ (B) . . . . (1).We recall the important conclusion:One can solve the Schrödinger equation SE via the statistical chains of the matrix Q without needing neither the Schrödinger equation itself, nor the method of separation of the variables x,y,z,t nor Newton's finite difference method (FDM).In Answers I and II, we solved the time-independent SE for the 1D and 2D infinite potential well cases.In this answer, we focus on the statistical solution of the Q matrix for time-independent SE for the case of a 3D closed box with infinite potential outside, which implies that the wave function Ψ = 0 outside the box.Case A:A 3D cube divided into 27 equidistant states (nodes)The statistical transition of the 27x27 real matrix B is given by,0.00 0.1667 0.00 0.1667 0.00 0.00 0.00 0.00 0.00 0.1667 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.1667 0.00 0.1667 0.00 0.1667 0.00 0.00 0.00 0.00 0.00 0.1667 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.1667 0.00 0.00 0.00 0.1667 0.00 0.00 0.00 0.00 0.00 0.1667 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.1667 0.00 0.00 0.00 0.1667 0.00 0.1667 0.00 0.00 0.00 0.00 0.00 0.1667 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.1667 0.00 0.1667 0.00 0.1667 0.00 0.1667 0.00 0.00 0.00 0.00 . . . etc.
The corresponding Q 27x27 complex transition matrix is given by equation 1 but cannot be presented here because its large size exceeds the frame of a page.
We therefore move on to the simpler case.
Case B:
A 3D cube of 8 equidistant states (vertices)
The statistical transition of the real 8x8 matrix B is given by,
0 1/6 0 1/6 1/6 0 0 0
1/6 0 1/6 0 0 1/6 0 0
0 1/6 0 1/6 0 0 1/6 0
1/6 0 1/6 0 0 0 0 1/6
1/6 0 0 0 0 1/6 0 1/6
0 1/6 0 0 1/6 0 1/6 0
0 0 1/6 0 0 1/6 0 1/6
0 0 0 1/6 1/6 0 1/6 0
And the corresponding Q 8x8 complex transition matrix is given by Equation 1 and is shown below as follows,
Matrix Q 8x8
((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48
((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16
((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48
((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48
((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48
((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48
((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48
((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16
**
**The destination transfer matrix W = 1(I-Q) will be
**
((105+35i)*2^0.5+(63+45i)*6^0.5+928)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5-(63-45i)*6^0.5+32)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840
((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5+(63+45i)*6^0.5+928)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5-(63-45i)*6^0.5+32)/840
((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5+(63+45i)*6^0.5+928)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105-35i)*2^0.5-(63-45i)*6^0.5+32)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840
((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5+(63+45i)*6^0.5+928)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5-(63-45i)*6^0.5+32)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840
((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5-(63-45i)*6^0.5+32)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105+35i)*2^0.5+(63+45i)*6^0.5+928)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840
((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5-(63-45i)*6^0.5+32)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5+(63+45i)*6^0.5+928)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840
((105-35i)*2^0.5-(63-45i)*6^0.5+32)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5+(63+45i)*6^0.5+928)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840
((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5-(63-45i)*6^0.5+32)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5-(21+15i)*6^0.5+64)/840 ((105-35i)*2^0.5+(21-15i)*6^0.5+176)/840 ((105+35i)*2^0.5+(63+45i)*6^0.5+928)/840
The final step is to calculate the eigenvalues of the energy on the 8 vertices of the box enclosed in an infinite exterior potential by multiplying the complex matrix W by the potential vector V.
I-First case for a uniform potential V given by the vector, [1,1,1,1,1,1,1,1 units]T .
The numerical result of the REAL energy eigenvalues is,
2^0.5+2 3.14
2^0.5+2 3.14
2^0.5+2 3.14
2^0.5+2 3.14
2^0.5+2 which is = 3.14
2^0.5+2 3.14
2^0.5+2 3.14
2^0.5+2 3.14
II-Second case for a non-uniform potential V given by the vector, [11,12,13,14,15,16,17,18 units]T .
The numerical result of the solution for the energy eigenvalues is,
(44.0 - 0.175i)
(44.865+0.175i)
(47.388-0.175i)
(48.245+0.175i)
(50.767-0.175i)
(51.625+0.175i)
(54.147-0.175i)
(55.004+0.175i)
all of which are almost real.
The conclusion is that:
i-√2 appears in the 1D, 2D and 3D energy distribution following equation 1.
ii- the transition matrix B and the transfer matrix W are complex but the resulting energy eigenvalues for the real external potential should be all be real.
*To be continued.
Answer IV- continued
In this answer, we explain the numerical statistical solution to the time-dependent SE without needing the SE itself in the same way used for solve the transient heat equation without using the heat equation.
In other words, we completely ignore the SE and the Bohr/Copenhagen interpretation as if they never existed.
The numerical statistical solution to SE will be,
Ψ=W . b. . . ..(1)
where b is the vector of the real potential applied to the quantum particle and W is the complex quantum transfer matrix expressed by:
W=Q + Q^2+Q^3+ . . .Q^N . . . . (2)
It was shown before that,
Q=Sqrt (B).. . . (3)
where B is the well-known real transition matrix used to find time-dependent solutions for the Poisson, Laplace, and heat diffusion partial differential equations.
Equations 1,2 show that the solutions of Schrödinger's equation depend only on the shape of the potential and the boundary conditions as expected.
In order not to worry too much about the details of the theory, let's move on to the following illustrative specific application without loss of generality:
Consider the simplest Cartesian geometrical shape , a cube of length L and eight vertices that represent the nodes or quantum states.
The B 8x8 transition matrix is given by,
0 1/6 0 1/6 1/6 0 0 0
1/6 0 1/6 0 0 1/6 0 0
0 1/6 0 1/6 0 0 1/6 0
1/6 0 1/6 0 0 0 0 1/6
1/6 0 0 0 0 1/6 0 1/6
0 1/6 0 0 1/6 0 1/6 0
0 0 1/6 0 0 1/6 0 1/6
0 0 0 1/6 1/6 0 1/6 0
and the complex quantum transition matrix Q=Sqrt (B) would be,
[Note the appearance of 2^1/2 which has a special importance in quantum mechanics.]((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+ 3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3-3i)*2 ^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-( 1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3- 3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3+3i)*2 ^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-( 1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+ i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1-i)*2 ^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+( 1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3- 3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3+3i)*2 ^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-( 1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1- i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1+i)*2 ^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-( 1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3+ 3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3-3i)*2 ^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16 ((3-3i)*2^0.5+( 1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3- 3i)*2^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3+3i)*2 ^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((1+i)*2^0.5+( 1+i)*6^0.5)/16 ((3-3i)*2^0.5+(1-i)*6^0.5)/48((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((1-i)*2^0.5-(1-i)*6^0.5)/16 ((3+ 3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+(1-i)*6^0.5)/48 ((3-3i)*2 ^0.5+(1-i)*6^0.5)/48 ((3+3i)*2^0.5-(1+i)*6^0.5)/48 ((3-3i)*2^0.5+( 1-i)*6^0.5)/48 ((1+i)*2^0.5+(1+i)*6^0.5)/16
It can be shown that equation 2 for a sufficiently large number of time jumps or iterations N which gives the stationary time-independent solution reduces to,
W= 1/ (I-Q) . . . . .(4)
Therefore the transfer matrix W is given by,
((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840
((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840
((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840
((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840
((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840
((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840
((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840
((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840
And the steady-state numerical solution Eps=(W-I) . b for the vector of unit boundary conditions,
b= [1,1,1,1,1,1,1,1] T, becomes,
2^0.5+1
2^0.5+1
2^0.5+1
2^0.5+1
2^0.5+1
2^0.5+1
2^0.5+1
2^0.5+1
where all eigenvalues of energy are real as expected.
[Note again the multiple appearance of 2^1/2 which has special significance in quantum mechanics.]
To be continued.
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