Question with superposition principle. [Archive]

Benjamin Schulz

Apr8-04, 06:38 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hello group.\n\nI\'m an undergraduate student from Germany\n\nI\'ve a question regarding the superposition principle.\n\nThe schroedinger equation is linear, so if \\psi_{1} is a solution and\n\\psi_{2} too, then \\Psi=\\psi_{1}+\\psi_{2} is a solution.\n\nNow suppose we have a Schroedinger equation of an N electron system.\nThis system may be a cable which is connected to a battery. That means\nthe many particle system is in a state that carries electric current.\nMoving electrons create a magnetic field. But due to the nature of the\nelectrons they react under magnetic fields. In formulas:\n\nWe have now the many particle equation under the field with Hamiltonian:\n\nH=\\sum_{j=0}^{N}{(p_j-eA(r_j))/(2m_0)}+ 0.5\\sum_{i,j,i\\neqj} {V(|r_j-r_i|))}\n\nwhich is the valid Hamiltonian that acts on the many particle density\nfunction \\Psi which itself depends from N space coordinates and the time.\n\nThe Hamiltonian contains the expression A. But since A is depends from\nthe magnetic field created by the current, it contains a contribution\n\nA\'(r_j) that depends on\n\n\\int \\frac{ j_e}{ (|r_j-r\'|)}dr\'\n\nwhere j_e is the current density\n\nj_e=e\\hbar/(2im_0)N\\int(\\Psi*\\nabla\\Psi))+e^2N/m_0\\int|\\Psi|^2A(r)\n\n\nthis would mean that H is state dependent.\n\nDoesn\'t this invalidate the superposition principle?\n\nWhat do physicists say about that?\n\nthank you for answers.\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello group.

I'm an undergraduate student from Germany

I've a question regarding the superposition principle.

The schroedinger equation is linear, so if \psi_{1} is a solution and
\psi_{2} too, then \Psi=\psi_{1}+\psi_{2} is a solution.

Now suppose we have a Schroedinger equation of an N electron system.
This system may be a cable which is connected to a battery. That means
the many particle system is in a state that carries electric current.
Moving electrons create a magnetic field. But due to the nature of the
electrons they react under magnetic fields. In formulas:

We have now the many particle equation under the field with Hamiltonian:

H=\sum_{j=0}^{N}{(p_j-eA(r_j))/(2m_0)}+[/itex] .5\sum_{i,j,i\neqj} {V(|r_j-r_i|))}

which is the valid Hamiltonian that acts on the many particle density
function \Psi which itself depends from N space coordinates and the time.

The Hamiltonian contains the expression A. But since A is depends from
the magnetic field created by the current, it contains a contribution

A'(r_j) that depends on

[itex]\int \frac{ j_e}{ (|r_j-r'|)}dr'

where j_e is the current density

j_e=e\hbar/(2im_0)N\int(\Psi*\nabla\Psi))+e^{2N}/m_0\int|\Psi|^2A(r)

this would mean that H is state dependent.

Doesn't this invalidate the superposition principle?

What do physicists say about that?

thank you for answers.

Igor Khavkine

Apr11-04, 11:44 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nBenjamin Schulz <Benjamin_Schulz@gmx.de> wrote in message news:<c51h5i\\$lpq\\$00\\$1@news.t-online.com>...\n\n> We have now the many particle equation under the field with Hamiltonian:\n>\n> H=\\sum_{j=0}^{N}{(p_j-eA(r_j))/(2m_0)}+ 0.5\\sum_{i,j,i\\neqj} {V(|r_j-r_i|))}\n>\n> which is the valid Hamiltonian that acts on the many particle density\n> function \\Psi which itself depends from N space coordinates and the time.\n>\n> The Hamiltonian contains the expression A. But since A is depends from\n> the magnetic field created by the current, it contains a contribution\n>\n> A\'(r_j) that depends on\n>\n> \\int \\frac{ j_e}{ (|r_j-r\'|)}dr\'\n>\n> where j_e is the current density\n>\n> j_e=e\\hbar/(2im_0)N\\int(\\Psi*\\nabla\\Psi))+e^2N/m_0\\int|\\Psi|^2A(r)\n>\n> this would mean that H is state dependent.\n>\n> Doesn\'t this invalidate the superposition principle?\n\nWhat you are interested in is the electromagnetic field generated\nby quantum charged particles. In this case you must also treat\nthe electromagnetic field quantum mechanically (unless you are\nmaking approximations). This is where quantization of the EM\nfield comes in. Once both the electrons and photons are quantized\nyou can have a consistent theory.\n\nIgor\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Benjamin Schulz <Benjamin_Schulz@gmx.de> wrote in message news:<c51h5i$lpq$00$1@news.t-online.com>...

> We have now the many particle equation under the field with Hamiltonian:
>
> H=\sum_{j=0}^{N}{(p_j-eA(r_j))/(2m_0)}+ .5\sum_{i,j,i\neqj} {V(|r_j-r_i|))}
>
> which is the valid Hamiltonian that acts on the many particle density
> function \Psi which itself depends from N space coordinates and the time.
>
> The Hamiltonian contains the expression A. But since A is depends from
> the magnetic field created by the current, it contains a contribution
>
> A'(r_j) that depends on
>
> \int \frac{ j_e}{ (|r_j-r'|)}dr'
>
> where j_e is the current density
>
> j_e=e\hbar/(2im_0)N\int(\Psi*\nabla\Psi))+e^{2N}/m_0\int|\Psi|^2A(r)
>
> this would mean that H is state dependent.
>
> Doesn't this invalidate the superposition principle?

What you are interested in is the electromagnetic field generated
by quantum charged particles. In this case you must also treat
the electromagnetic field quantum mechanically (unless you are
making approximations). This is where quantization of the EM
field comes in. Once both the electrons and photons are quantized
you can have a consistent theory.

Igor

William R. Frensley

Apr11-04, 11:44 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nBenjamin Schulz wrote:\n....\n> I\'ve a question regarding the superposition principle.\n>\n> The schroedinger equation is linear, so if \\psi_{1} is a solution and\n> \\psi_{2} too, then \\Psi=\\psi_{1}+\\psi_{2} is a solution.\n>\n> Now suppose we have a Schroedinger equation of an N electron system.\n> This system may be a cable which is connected to a battery. That means\n> the many particle system is in a state that carries electric current.\n> Moving electrons create a magnetic field. But due to the nature of the\n> electrons they react under magnetic fields. In formulas:\n......\n>\n> The Hamiltonian contains the expression A. But since A is depends from\n> the magnetic field created by the current, it contains a contribution\n>\n> A\'(r_j) that depends on\n>\n> \\int \\frac{ j_e}{ (|r_j-r\'|)}dr\'\n>\n> where j_e is the current density\n>\n> j_e=e\\hbar/(2im_0)N\\int(\\Psi*\\nabla\\Psi))+e^2N/m_0\\int|\\Psi|^2A(r)\n>\n>\n> this would mean that H is state dependent.\n>\n> Doesn\'t this invalidate the superposition principle?\n>\nThis sort of nonlinearity occurs whenever we approximate a multi-particle\nproblem with an averaged or effective-field approach. It appears in the\neven simpler case of electrostatic screening, as in the Hartree approximation\nfor a multi-electron atom. If we were to write out the full multi-particle\nSchroedinger equation, (which is expressed in terms of a single wavefunction\nthat depends upon all the coordinates of all the particles), we would have an\nequation which obeys superposition. The development of nonlinearity is the\nresult of "hiding" many of the details of the problem.\n\n- Bill Frensley\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Benjamin Schulz wrote:
....
> I've a question regarding the superposition principle.
>
> The schroedinger equation is linear, so if \psi_{1} is a solution and
> \psi_{2} too, then \Psi=\psi_{1}+\psi_{2} is a solution.
>
> Now suppose we have a Schroedinger equation of an N electron system.
> This system may be a cable which is connected to a battery. That means
> the many particle system is in a state that carries electric current.
> Moving electrons create a magnetic field. But due to the nature of the
> electrons they react under magnetic fields. In formulas:
......
>
> The Hamiltonian contains the expression A. But since A is depends from
> the magnetic field created by the current, it contains a contribution
>
> A'(r_j) that depends on
>
> \int \frac{ j_e}{ (|r_j-r'|)}dr'
>
> where j_e is the current density
>
> j_e=e\hbar/(2im_0)N\int(\Psi*\nabla\Psi))+e^{2N}/m_0\int|\Psi|^2A(r)
>
>
> this would mean that H is state dependent.
>
> Doesn't this invalidate the superposition principle?
>
This sort of nonlinearity occurs whenever we approximate a multi-particle
problem with an averaged or effective-field approach. It appears in the
even simpler case of electrostatic screening, as in the Hartree approximation
for a multi-electron atom. If we were to write out the full multi-particle
Schroedinger equation, (which is expressed in terms of a single wavefunction
that depends upon all the coordinates of all the particles), we would have an
equation which obeys superposition. The development of nonlinearity is the
result of "hiding" many of the details of the problem.

- Bill Frensley

Hendrik van Hees

Apr11-04, 11:44 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nBenjamin Schulz wrote:\n\n> H=\\sum_{j=0}^{N}{(p_j-eA(r_j))/(2m_0)}+ 0.5\\sum_{i,j,i\\neqj}\n> {V(|r_j-r_i|))}\n>\n> which is the valid Hamiltonian that acts on the many particle density\n> function \\Psi which itself depends from N space coordinates and the\n> time.\n>\n> The Hamiltonian contains the expression A. But since A is depends from\n> the magnetic field created by the current, it contains a contribution\n>\n> A\'(r_j) that depends on\n>\n> \\int \\frac{ j_e}{ (|r_j-r\'|)}dr\'\n>\n> where j_e is the current density\n>\n> j_e=e\\hbar/(2im_0)N\\int(\\Psi*\\nabla\\Psi))+e^2N/m_0\\int|\\Psi|^2A(r)\n>\n>\n> this would mean that H is state dependent.\n>\n> Doesn\'t this invalidate the superposition principle?\n>\n> What do physicists say about that?\n>\n> thank you for answers.\n\nWhat you described here is an age old problem which is imho not solved\ncompletely yet, namely the reaction of charged particles to their own\nradiation field.\n\nThe trouble is a little bit tamed in quantum theory compared to\nclassical electron theory (i.e. Maxwell-Lorentz theory of charged point\nparticles).\n\nIn this case the quantum case the most efficient way is to use quantum\nfield theory to describe particles and fields on a common basis.\n\nWe start with vacuum quantum field theory. There it turns out that you\nhave a Hamiltonian, where particles and fields are described by quantum\nfields, i.e., field operators, acting on a Fock space (built as the\nsuperposition of symmetrised (bosons) or anti-symmetrised (fermions)\nproducts of one-particle states, and taking the orthogonal sum of the\nspaces for each number of particles (N=0=vacuum, N=1, etc.).\n\nNow you start with perturbation theory, for instance in the case of QED,\nyou take the free electron-positron field (a Dirac field) and the free\nelectromagnetic (photon) field (a massless helicity 1 and a helicity\n-1-field together, in order to build a parity invariant Hamiltonian,\nbecause the electromagnetic interaction is known to conserve parity)\nand do perturbation theory for the coupling of the electron-positron\nfield to the photon field.\n\nIt turns out that you can neatly organise the perturbation series in\nterms of Feynman diagrams. The Feynman diagrams in higher order\nperturbation theory (in QED it\'s starting from 2nd order) can contain\nloops, which correspond to the interaction of electrons with its own\nelectromagnetic field.\n\nThe most simple example is the self-energy diagram, which corresponds to\na renormalisation of the wave function and a mass shift (this should be\nsomehow familiar from the old-fasioned perturbation theory treated in\nthe introductary course on quantum mechanics, where in time-independent\nperturbation theory, you calculate corrections to the energy and to the\ncorresponding energy eigen states in a formal series with respect to a\nsmall parameter in the perturbative part of the Hamiltonian).\n\nIn relativistic qft it turns out that the corresponding expression is\ndivergent, as in the analogous case of classical Maxwell Lorentz\ntheory, but it is much less difficult than there, because in the\nquantum case one can systematically shift the infinities to the\nunobservable "bare quantitities", so that all the quantities of QED are\nperfectly finite (the wave function normalisation, the mass of the\nelectron and the charge of the electron or fine structure constant). In\nthe classical theory, this is not possible at any order perturbation\ntheory.\n\nThis is, what I meant in the beginning, when I wrote that this problem\nis partially solved in quantum field theory.\n\nYou example is of course much more complicated, because you deal with a\nreal many-body problem, namely many electrons in a wire. This can be\ntreated also with help of quantum field theory at finite temperatures\nor general non-equilibrium quantum field theory.\n\n--\nHendrik van Hees Cyclotron Institute\nPhone: +1 979/845-1411 Texas A&M University\nFax: +1 979/845-1899 Cyclotron Institute, MS-3366\nhttp://theory.gsi.de/~vanhees/ College Station, TX 77843-3366\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Benjamin Schulz wrote:

> H=\sum_{j=0}^{N}{(p_j-eA(r_j))/(2m_0)}+ .5\sum_{i,j,i\neqj}
> {V(|r_j-r_i|))}
>
> which is the valid Hamiltonian that acts on the many particle density
> function \Psi which itself depends from N space coordinates and the
> time.
>
> The Hamiltonian contains the expression A. But since A is depends from
> the magnetic field created by the current, it contains a contribution
>
> A'(r_j) that depends on
>
> \int \frac{ j_e}{ (|r_j-r'|)}dr'
>
> where j_e is the current density
>
> j_e=e\hbar/(2im_0)N\int(\Psi*\nabla\Psi))+e^{2N}/m_0\int|\Psi|^2A(r)
>
>
> this would mean that H is state dependent.
>
> Doesn't this invalidate the superposition principle?
>
> What do physicists say about that?
>
> thank you for answers.

What you described here is an age old problem which is imho not solved
completely yet, namely the reaction of charged particles to their own
radiation field.

The trouble is a little bit tamed in quantum theory compared to
classical electron theory (i.e. Maxwell-Lorentz theory of charged point
particles).

In this case the quantum case the most efficient way is to use quantum
field theory to describe particles and fields on a common basis.

We start with vacuum quantum field theory. There it turns out that you
have a Hamiltonian, where particles and fields are described by quantum
fields, i.e., field operators, acting on a Fock space (built as the
superposition of symmetrised (bosons) or anti-symmetrised (fermions)
products of one-particle states, and taking the orthogonal sum of the
spaces for each number of particles (N=0=vacuum, N=1, etc.).

Now you start with perturbation theory, for instance in the case of QED,
you take the free electron-positron field (a Dirac field) and the free
electromagnetic (photon) field (a massless helicity 1 and a helicity
-1-field together, in order to build a parity invariant Hamiltonian,
because the electromagnetic interaction is known to conserve parity)
and do perturbation theory for the coupling of the electron-positron
field to the photon field.

It turns out that you can neatly organise the perturbation series in
terms of Feynman diagrams. The Feynman diagrams in higher order
perturbation theory (in QED it's starting from 2nd order) can contain
loops, which correspond to the interaction of electrons with its own
electromagnetic field.

The most simple example is the self-energy diagram, which corresponds to
a renormalisation of the wave function and a mass shift (this should be
somehow familiar from the old-fasioned perturbation theory treated in
the introductary course on quantum mechanics, where in time-independent
perturbation theory, you calculate corrections to the energy and to the
corresponding energy eigen states in a formal series with respect to a
small parameter in the perturbative part of the Hamiltonian).

In relativistic qft it turns out that the corresponding expression is
divergent, as in the analogous case of classical Maxwell Lorentz
theory, but it is much less difficult than there, because in the
quantum case one can systematically shift the infinities to the
unobservable "bare quantitities", so that all the quantities of QED are
perfectly finite (the wave function normalisation, the mass of the
electron and the charge of the electron or fine structure constant). In
the classical theory, this is not possible at any order perturbation
theory.

This is, what I meant in the beginning, when I wrote that this problem
is partially solved in quantum field theory.

You example is of course much more complicated, because you deal with a
real many-body problem, namely many electrons in a wire. This can be
treated also with help of quantum field theory at finite temperatures
or general non-equilibrium quantum field theory.

--
Hendrik van Hees Cyclotron Institute
Phone: +1 979/845-1411 Texas A&M University
Fax: +1 979/845-1899 Cyclotron Institute, MS-3366
http://theory.gsi.de/~vanhees/ College Station, TX 77843-3366