Complex Schrodinger Equation, references?

[dimachka]

Complex Schrödinger Equation, references??

hopefully i can explain what i am looking for well enough for somebody to understand. I am interested in finding any references for work that has been done on solving the Schrödinger equation in C^n rather than in R^n, as in on the complex plane with complex valued argument rather than the real case which is what we normally consider when doing quantum mechanics. Obviously i understand that a particle's probability to be located at the position (1 + i) is not exactly physically understandable. But nonetheless, I am interested in what work has been done on the Schrödinger equation in the complex plane and am hoping that somebody will be able to provide some references and/or ideas about the subject. (Also there will be major problems with any sort of normalization, but nonetheless I'm interested. )

I have tried minimal searching with google scholar and my universities article search but have not been able to find any references. Thank you for any input or help, i appreciate it greatly.

 

[masudr]

The Schrödinger equation is

$$\hat{H}\left|\psi\left(t\right)\right\rangle = i\hbar\frac{d}{dt}\left|\psi\left(t\right)\right\rangle.$$

For easier notation, I will now take $\left|\psi\left(t\right)\right\rangle$ at some moment $t=t_0$ so that $\left|\psi\right\rangle=\left|\psi\left(t_0\right)\right\rangle.$ You can choose then to express $|\psi\rangle$ in the $|x\rangle$ basis and get what we call the wavefunction. If the state space was representing a system with $n$ spatial degrees of freedom, then we get a function defined on $\mathbb{R}^n:$

$$\langle x^1, x^2...x^n|\psi\rangle = \psi\left(x^1,x^2...x^n\right).$$

If you want to, you can express $|\psi\rangle$ in any basis you want, including ones that are defined on complex numbers.

N.B. the superscripts on my position variables are representing co-ordinate numbering, not powers.

------

If you mean, on the other hand, the energy eigenvalue equation (which is often promoted to being called the time independent Schrödinger equation, which I think is silly really, because it's equivalent to the momentum eignevalue equation, or the angular momentum eigenvalue equation &c.):

$$\hat{H}\left|\psi\right\rangle=E\left|\psi\right\rangle,$$

then you can also cast that into any basis you prefer, as long as you remember that the Hamiltonian will also have to be in that basis.

 

[vanesch]

hopefully i can explain what i am looking for well enough for somebody to understand. I am interested in finding any references for work that has been done on solving the Schrödinger equation in C^n rather than in R^n, as in on the complex plane with complex valued argument rather than the real case which is what we normally consider when doing quantum mechanics.

Ah, it is not so much the "Schroedinger equation" but the Hilbert space you are changing when doing so. In other words, you now have a position basis spanning the Hilbert space which is |x,y,z> with x,y, z complex instead of real, is that it ?

I can look upon this in two ways: or there is a requirement of analycity as a function of x, y and z, in which case, I don't think that there is any difference (the "real" solution being analytically continued in the complex x,y and z plane). Or, there is no such requirement, in which case the real and imaginary parts "lead their own life, and then I fail to see the difference with a 2-particle situation |x,y,z,u,v,w> relabelled:
|x + iu, y+iv,z+iw>.

(Also there will be major problems with any sort of normalization, but nonetheless I'm interested. )

Yes :-) An analytic function over the whole plane is constant, or unbounded...
Imagine the "real" plane wave exp(-i k x) with imaginary x...

On the other hand, this is exactly what is done for the TIME variable. There's lots of discussion about that (from Wick rotations, to statistical mechanics)

 

[dimachka]

Ah, it is not so much the "Schroedinger equation" but the Hilbert space you are changing when doing so. In other words, you now have a position basis spanning the Hilbert space which is |x,y,z> with x,y, z complex instead of real, is that it ?

I can look upon this in two ways: or there is a requirement of analycity as a function of x, y and z, in which case, I don't think that there is any difference (the "real" solution being analytically continued in the complex x,y and z plane). Or, there is no such requirement, in which case the real and imaginary parts "lead their own life, and then I fail to see the difference with a 2-particle situation |x,y,z,u,v,w> relabelled:
|x + iu, y+iv,z+iw>.



Yes :-) An analytic function over the whole plane is constant, or unbounded...
Imagine the "real" plane wave exp(-i k x) with imaginary x...

On the other hand, this is exactly what is done for the TIME variable. There's lots of discussion about that (from Wick rotations, to statistical mechanics)

I guess i should explain some more, as i did not explain well what i meant. I am interested in finding information on solving the general equation (-Y'' + V(z)*Y = k*dY/dt) where ' indicates spatial derivative, k is a constant, V is an arbitrary holomorphic function and Y is a function of the complex variable z and time. This is identical in form to the Schrödinger equation but I'm not too interested in normalization. Specifically, i don't want bounded solutions. I am not sure, but I think that the fact that V is a complex valued function mean the solutions are not just analytic continuations of real line solutions. Seperating the real and imaginary part of V, can yield two equations but I don't know what that implies since it yields two different equations.

Please feel free to tell me anything i am saying is irrelevant slash incompetent. I'm just interested in what input i can get, thanks a bunch!

 

[vanesch]

I guess i should explain some more, as i did not explain well what i meant. I am interested in finding information on solving the general equation (-Y'' + V(z)*Y = k*dY/dt) where ' indicates spatial derivative, k is a constant, V is an arbitrary holomorphic function and Y is a function of the complex variable z and time.

Ah, this is where you got me: V can be a complex function. Indeed, then the problem is different from the standard Schroedinger equation. Can't help much here...

 

[reilly]

In Ince's Ordinary Differential Equations (Dover Books) you'll find a long chapter on linear differential equations in the complex plane. But, as vanesch points out, there are problems with V(z) -- even if you restrict V to be real on the real axis.
Regards,
Reilly Atkinson

 

[satyadhar_joshi]

can u give some links to the free source things