View Full Version : Classical tunneling
Gentil Correa
May1-04, 07:53 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>The problem is well known: a ray of light propagating in a slab of\nglass reaches its (plane) boundary and enters vacuum. Above a certain\nangle of incidence, total reflection occurs. If, however, another\nsimilar slab, parallel to the first, is\nvery close to it, so that a very thin vaccum gap is left between them,\nit is seen that the ray reemerges in the other slab, although with a\nmuch weaker intensity. In the context of geometrical optics this is\nimpossible. However, the phenomenon exists, and can be derived in the\ncontext of wave optics. My question is a pedagogical one: you can find\nderivations of this "tunneling" in several textbooks, like the\nexcellent Sommerfeld "Optics", but the derivation involves a subtle\nuse of complex numbers, which seems suspicious to most undergraduate\nstudents. What I am looking for is a clear-cut derivation of this fact\nusing Maxwell equations directly, and without the trick of analytic\ncontinuation. Can someone help me?\n\nThanks\n\nGentil\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>The problem is well known: a ray of light propagating in a slab of
glass reaches its (plane) boundary and enters vacuum. Above a certain
angle of incidence, total reflection occurs. If, however, another
similar slab, parallel to the first, is
very close to it, so that a very thin vaccum gap is left between them,
it is seen that the ray reemerges in the other slab, although with a
much weaker intensity. In the context of geometrical optics this is
impossible. However, the phenomenon exists, and can be derived in the
context of wave optics. My question is a pedagogical one: you can find
derivations of this "tunneling" in several textbooks, like the
excellent Sommerfeld "Optics", but the derivation involves a subtle
use of complex numbers, which seems suspicious to most undergraduate
students. What I am looking for is a clear-cut derivation of this fact
using Maxwell equations directly, and without the trick of analytic
continuation. Can someone help me?
Thanks
Gentil
<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>gentil382001@hotmail.com (Gentil Correa) wrote in message news:<2c3b8202.0405010407.51bfae5b@posting.google. com>...\nbut the derivation involves a subtle\n> use of complex numbers, which seems suspicious to most undergraduate\n> students.\n\nWhy should complex numbers seem "suspicious" to anyone? If your\nstudents are for some reason uncomfortable with complex numbers that\'s\na much more serious problem than finding an alternative derivation of\nthis obscure effect.\n\nDavid\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>gentil382001@hotmail.com (Gentil Correa) wrote in message news:<2c3b8202.0405010407.51bfae5b@posting.google.com>...
but the derivation involves a subtle
> use of complex numbers, which seems suspicious to most undergraduate
> students.
Why should complex numbers seem "suspicious" to anyone? If your
students are for some reason uncomfortable with complex numbers that's
a much more serious problem than finding an alternative derivation of
this obscure effect.
David
Gentil Correa
May6-04, 07:52 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\nulmo@cheerful.com (Ulmo) wrote in message news:<53ca460a.0405021537.578028b2@posting.google. com>...\n> gentil382001@hotmail.com (Gentil Correa) wrote in message news:<2c3b8202.0405010407.51bfae5b@posting.google. com>...\n> but the derivation involves a subtle\n> > use of complex numbers, which seems suspicious to most undergraduate\n> > students.\n>\n> Why should complex numbers seem "suspicious" to anyone? If your\n> students are for some reason uncomfortable with complex numbers that\'s\n> a much more serious problem than finding an alternative derivation of\n> this obscure effect.\n>\n> David\n\nOf course, complex numbers are not "suspicious" to my students. They\nare sophomores and can handle complex algebra and the usual techniques\nof replacing sines and/or cosines by an exponential with imaginary\nargument.\nIt is analytic continuation (the "subtle use") which poses a problem,\nas they, at that point of their mathematical education, didn\'t have a\nproper course of analytical functions of one complex variable.\n\nGentil\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>ulmo@cheerful.com (Ulmo) wrote in message news:<53ca460a.0405021537.578028b2@posting.google.com>...
> gentil382001@hotmail.com (Gentil Correa) wrote in message news:<2c3b8202.0405010407.51bfae5b@posting.google.com>...
> but the derivation involves a subtle
> > use of complex numbers, which seems suspicious to most undergraduate
> > students.
>
> Why should complex numbers seem "suspicious" to anyone? If your
> students are for some reason uncomfortable with complex numbers that's
> a much more serious problem than finding an alternative derivation of
> this obscure effect.
>
> David
Of course, complex numbers are not "suspicious" to my students. They
are sophomores and can handle complex algebra and the usual techniques
of replacing sines and/or cosines by an exponential with imaginary
argument.
It is analytic continuation (the "subtle use") which poses a problem,
as they, at that point of their mathematical education, didn't have a
proper course of analytical functions of one complex variable.
Gentil
Rob Woodside
May7-04, 03:34 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>gentil382001@hotmail.com (Gentil Correa) wrote in message news:<2c3b8202.0405040216.5cb474a0@posting.google. com>...\nsnip\n> Of course, complex numbers are not "suspicious" to my students. They\n> are sophomores and can handle complex algebra and the usual techniques\n> of replacing sines and/or cosines by an exponential with imaginary\n> argument.\n> It is analytic continuation (the "subtle use") which poses a problem,\n> as they, at that point of their mathematical education, didn\'t have a\n> proper course of analytical functions of one complex variable.\n\nI vaguely remember from my sophomore year that Fresnel\'s evanescent\nwave is purely imaginary. Is that true??? It seems not, as that would\ngive the wave a negative energy density. Further, it would violate\nEinstein\'s remark to Born, when he was struggling with an\ninterpretation for the wave function, that everything would be all\nright if the square of the electric field was proportional to the\nprobability density for finding a photon.\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>gentil382001@hotmail.com (Gentil Correa) wrote in message news:<2c3b8202.0405040216.5cb474a0@posting.google.com>...
snip
> Of course, complex numbers are not "suspicious" to my students. They
> are sophomores and can handle complex algebra and the usual techniques
> of replacing sines and/or cosines by an exponential with imaginary
> argument.
> It is analytic continuation (the "subtle use") which poses a problem,
> as they, at that point of their mathematical education, didn't have a
> proper course of analytical functions of one complex variable.
I vaguely remember from my sophomore year that Fresnel's evanescent
wave is purely imaginary. Is that true??? It seems not, as that would
give the wave a negative energy density. Further, it would violate
Einstein's remark to Born, when he was struggling with an
interpretation for the wave function, that everything would be all
right if the square of the electric field was proportional to the
probability density for finding a photon.
Aaron Denney
May12-04, 01:52 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>On 2004-05-07, Rob Woodside <rwmw@telus.net> wrote:\n> gentil382001@hotmail.com (Gentil Correa) wrote in message news:<2c3b8202.0405040216.5cb474a0@posting.google. com>...\n> snip\n>> Of course, complex numbers are not "suspicious" to my students. They\n>> are sophomores and can handle complex algebra and the usual techniques\n>> of replacing sines and/or cosines by an exponential with imaginary\n>> argument.\n>> It is analytic continuation (the "subtle use") which poses a problem,\n>> as they, at that point of their mathematical education, didn\'t have a\n>> proper course of analytical functions of one complex variable.\n>\n> I vaguely remember from my sophomore year that Fresnel\'s evanescent\n> wave is purely imaginary. Is that true???\n\nIt\'s not. You can think of the "wave-vector" (the k in sin (k.x)) as being\npurely imaginary though, if you do the "express E-field as real part\nand B-field as imaginary part of e^{ik.x}". It oscillates in time, but\nexponentially decays in space as it gets further from the barrier.\n\n> It seems not, as that would give the wave a negative energy density.\n> Further, it would violate Einstein\'s remark to Born, when he was\n> struggling with an interpretation for the wave function, that\n> everything would be all right if the square of the electric field was\n> proportional to the probability density for finding a photon.\n\nYes, see above. It\'s good that you can see that interpretation\ndoesn\'t make physical sense.\n\n--\nAaron Denney\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>On 2004-05-07, Rob Woodside <rwmw@telus.net> wrote:
> gentil382001@hotmail.com (Gentil Correa) wrote in message news:<2c3b8202.0405040216.5cb474a0@posting.google.com>...
> snip
>> Of course, complex numbers are not "suspicious" to my students. They
>> are sophomores and can handle complex algebra and the usual techniques
>> of replacing sines and/or cosines by an exponential with imaginary
>> argument.
>> It is analytic continuation (the "subtle use") which poses a problem,
>> as they, at that point of their mathematical education, didn't have a
>> proper course of analytical functions of one complex variable.
>
> I vaguely remember from my sophomore year that Fresnel's evanescent
> wave is purely imaginary. Is that true???
It's not. You can think of the "wave-vector" (the k in sin (k.x)) as being
purely imaginary though, if you do the "express E-field as real part
and B-field as imaginary part of e^{ik.x}". It oscillates in time, but
exponentially decays in space as it gets further from the barrier.
> It seems not, as that would give the wave a negative energy density.
> Further, it would violate Einstein's remark to Born, when he was
> struggling with an interpretation for the wave function, that
> everything would be all right if the square of the electric field was
> proportional to the probability density for finding a photon.
Yes, see above. It's good that you can see that interpretation
doesn't make physical sense.
--
Aaron Denney
-><-
Rob Woodside
May16-04, 12:58 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>Aaron Denney <wnoise@ofb.net> wrote in message news:<slrnc9sndu.3l7.wnoise@ofb.net>...\n> On 2004-05-07, Rob Woodside <rwmw@telus.net> wrote:\n> > I vaguely remember from my sophomore year that Fresnel\'s evanescent\n> > wave is purely imaginary. Is that true???\n>\n> It\'s not. You can think of the "wave-vector" (the k in sin (k.x)) as being\n> purely imaginary though, if you do the "express E-field as real part\n> and B-field as imaginary part of e^{ik.x}". It oscillates in time, but\n> exponentially decays in space as it gets further from the barrier.\n\nThanks Aaron. With the delays of posting etc. I looked this up and\nfound what you say. Perhaps this reference is useful for Gentil:\n\nG.L. Pollack and D.R. Stump, " Electromagnetism", pp485-505, Addison\nWesley, San Francisco, (2002)\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>Aaron Denney <wnoise@ofb.net> wrote in message news:<slrnc9sndu.3l7.wnoise@ofb.net>...
> On 2004-05-07, Rob Woodside <rwmw@telus.net> wrote:
> > I vaguely remember from my sophomore year that Fresnel's evanescent
> > wave is purely imaginary. Is that true???
>
> It's not. You can think of the "wave-vector" (the k in sin (k.x)) as being
> purely imaginary though, if you do the "express E-field as real part
> and B-field as imaginary part of e^{ik.x}". It oscillates in time, but
> exponentially decays in space as it gets further from the barrier.
Thanks Aaron. With the delays of posting etc. I looked this up and
found what you say. Perhaps this reference is useful for Gentil:
G.L. Pollack and D.R. Stump, " Electromagnetism", pp485-505, Addison
Wesley, San Francisco, (2002)
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