Ground state of a quantum system of identical boson particles
Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 (1) Laboratoire PhLAM, UMR CNRS 8523, Universit\303\251 Lille 1, F-59655, France(2) MaplesoftDeparting from the Energy of a quantum system of identical boson particles, the field equation is derived. This is the Gross-Pitaevskii equation (GPE). A continuity equation for this system is also derived, showing that the velocity flow satisfies 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, i.e.: is irrotational.
<Text-field style="Heading 1" layout="Heading 1">The Gross-Pitaevskii equation</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Problem: derive the field equation describing the ground state of a quantum system of identical particles (bosons), that is, the Gross-Pitaevskii equation (GPE).Background: The Gross-Pitaevskii equation is particularly useful to describe Bose Einstein condensates (BEC) of cold atomic gases [3, 4, 5], that is, an ensemble of identical quantum boson particles that interact with each other with an interaction constant G. The temperature of these cold atomic gases is typically in the \342\210\274100 nano-Kelvin range. The atom-atom interactions are repulsive for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= and attractive for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= (which could lead to some instabilities). The GPE is also widely used in non-linear optics to model the propagation of light in optical fibers. In this area, GPE is known as "non-linear Schr\303\266dinger equation", and the non-linearity comes from the Kerr effect [6].
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>One can derive this field equation the usual way, in two steps:
Construct the Lagrangian for the system, and with it write the action functionalThe Gross-Pitaevskii equation is obtained minimizing this action, i.e., equating to zero its functional derivative with respect to the boson field.Derivation: The system is assumed to be at sufficiently low temperature such that the particles all share the same quantum ground state 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkcHNpRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the particle-field function and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the total particle number: 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. To construct the Lagrangian we thus depart from the energy density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for a quantum system of identical boson particles.The version of Physics used is from November/26 (or later), available at the Maplesoft Physics Research & Development 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energy density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for a quantum system of identical boson particles is (see [3])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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NDYzO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the Planck constant divided by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y4Ri9GLw==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= the mass of a single particle, 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 a complex field, 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 an arbitrary external potential and the interaction term LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= takes into account atom-atom interactions. So set the real objects for this 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 Lagrangian density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is defined in terms of the Energy E in the usual 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The corresponding Action 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 the action gives the field equations, so taking the functional 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Equating this result to 0 is the desired field equation, GPE. This result can be compacted to arrive at the standard form of the GPE. For instance, in the right-hand side of we see a Laplacian in disguise. So take the conjugate and isolate the time derivative: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 in the 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The product LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRwc2lGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUkmbW92ZXJHRiQ2JkYxLUY7Ni1RLSZjb25qdWdhdGUwO0YnRjdGPkZAL0ZDUSV0cnVlRidGREZGRkhGSkZMRk9GSi9JK21zZW1hbnRpY3NHRiRRKmNvbmp1Z2F0ZUYnRjdGK0Y3 can be rewritten as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYoLUkjbWlHRiQ2JVEkcHNpRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y1L0krbXNlbWFudGljc0dGJFEkYWJzRicvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR0Y9RjhGNQ== and, collecting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkcHNpRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, we arrive at the standard form of the Gross\342\200\223Pitaevskii 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 the GPE looks like the usual Schr\303\266dinger equation, except there is a non-linear term acting as a potential proportional to the local field 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
<Text-field style="Heading 2" size="18" layout="Heading 2"><Font size="18">Continuity equation for a quantum system of identical particles</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Like for the standard Schr\303\266dinger equation, it is possible to derive a continuity equation for the ground state of a quantum system of identical particles that is similar to the one in fluid mechanics. Because the non linear term 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 is real, the continuity equation will be independent of this non linearity (and of the potential LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as well). To obtain the continuity equation, GPE is first multiplied by 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 ; then the complex conjugate of the resulting product is subtracted: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The left hand side of can be integrated, 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So 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The right-hand side of this result can also be rewritten as a divergence, 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Take now the divergence and compare with its expanded 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 by the proper factors of the right-hand side of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYoLUkjbW5HRiQ2JFEiaUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtSSNtaUdGJDYlUSkmaHNsYXNoO0YnLyUnaXRhbGljR1EldHJ1ZUYnL0Y1USdpdGFsaWNGJy1GODYuUSJ+RidGU0ZWRjtGPkZARkJGREZGRkhGSkZNLUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRaWptLy1JLCVEaXZlcmdlbmNlRzYiNiMsJiomLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjLUkkcHNpRzYiNiZJInhHJSpwcm90ZWN0ZWRHSSJ5RyUqcHJvdGVjdGVkR0kiekclKnByb3RlY3RlZEdJInRHNiIiIiItSSolR3JhZGllbnRHNiI2Iy1JJHBzaUc2IjYmSSJ4RyUqcHJvdGVjdGVkR0kieUclKnByb3RlY3RlZEdJInpHJSpwcm90ZWN0ZWRHSSJ0RzYiIiIiIiIiKiYtSSRwc2lHNiI2JkkieEclKnByb3RlY3RlZEdJInlHJSpwcm90ZWN0ZWRHSSJ6RyUqcHJvdGVjdGVkR0kidEc2IiIiIi1JKiVHcmFkaWVudEc2IjYjLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjLUkkcHNpRzYiNiZJInhHJSpwcm90ZWN0ZWRHSSJ5RyUqcHJvdGVjdGVkR0kiekclKnByb3RlY3RlZEdJInRHNiIiIiIhIiIsJiomLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjLUkkcHNpRzYiNiZJInhHJSpwcm90ZWN0ZWRHSSJ5RyUqcHJvdGVjdGVkR0kiekclKnByb3RlY3RlZEdJInRHNiIiIiItSSslTGFwbGFjaWFuRzYiNiMtSSRwc2lHNiI2JkkieEclKnByb3RlY3RlZEdJInlHJSpwcm90ZWN0ZWRHSSJ6RyUqcHJvdGVjdGVkR0kidEc2IiIiIiIiIiomLUkkcHNpRzYiNiZJInhHJSpwcm90ZWN0ZWRHSSJ5RyUqcHJvdGVjdGVkR0kiekclKnByb3RlY3RlZEdJInRHNiIiIiItSSslTGFwbGFjaWFuRzYiNiMtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiMtSSRwc2lHNiI2JkkieEclKnByb3RlY3RlZEdJInlHJSpwcm90ZWN0ZWRHSSJ6RyUqcHJvdGVjdGVkR0kidEc2IiIiIiEiIkYnLyUrYWN0aW9udHlwZUdRPG1hcGxlc29mdC5jb206bGFiZWwoTDE2NzQ4KUYnLyUldmlld0dRJmxhYmVsRictRlA2I1EhRictRiM2JS1GMTYkUSIyRidGNC1GODYtRlpGNEY7Rj5GQEZCRkRGRkZIRkpGTS1GUDYlUSJtRidGU0ZWLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZkcC8lKWJldmVsbGVkR0Y9RmJvThe right-hand side of this result is equal to the right-hand side of . So subtract this result and isolate the time derivative 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To express as a typical continuity equation, the argument of the divergence operator can be rewritten as the product of the particle density 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 times a velocity field 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, where the density satisfies 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. 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 the argument of the divergence 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 can be expressed 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Introducing now the particle density n and expandingLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEkYWJzRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRJFBzaUYnRi9GMkYyLUYjNiMtSSNtbkdGJDYkUSIyRidGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmduLUZKNi1RIn5GJ0YyRk1GT0ZRRlNGVUZXRlkvRmZuUSYwLjBlbUYnL0ZpbkZeby1GLDYlUSJuRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRjk2JC1GIzYpLUYsNiVRInhGJ0Zjb0Zlby1GSjYtUSIsRidGMkZNL0ZQRmRvRlFGU0ZVRldGWUZdby9GaW5RLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRmNvRmVvRl5wLUYsNiVRInpGJ0Zjb0Zlb0ZecC1GLDYlUSJ0RidGY29GZW9GMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiNRIUYnLUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRanQvKiQpLUkkYWJzRyUqcHJvdGVjdGVkRzYjLUkkcHNpRzYiNiZJInhHJSpwcm90ZWN0ZWRHSSJ5RyUqcHJvdGVjdGVkR0kiekclKnByb3RlY3RlZEdJInRHNiIiIiMiIiItSSJuRzYiNiZJInhHJSpwcm90ZWN0ZWRHSSJ5RyUqcHJvdGVjdGVkR0kiekclKnByb3RlY3RlZEdJInRHNiJGJy8lK2FjdGlvbnR5cGVHUTxtYXBsZXNvZnQuY29tOmxhYmVsKEwxNjc2OClGJy8lJXZpZXdHUSZsYWJlbEYnLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZSLyUqc3ltbWV0cmljR0ZSLyUobGFyZ2VvcEdGUi8lLm1vdmFibGVsaW1pdHNHRlIvJSdhY2NlbnRHRlIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GPjYlLUZBNiNRYGNsLy1JJiVkaWZmRzYiNiQqJCktSSRhYnNHJSpwcm90ZWN0ZWRHNiMtSSRwc2lHNiI2JkkieEclKnByb3RlY3RlZEdJInlHJSpwcm90ZWN0ZWRHSSJ6RyUqcHJvdGVjdGVkR0kidEc2IiIiIyIiIkkidEc2IioqKiYjIiIiIiIjIiIiXiMiIiIiIiIiIiJJJyZoYmFyO0c2IiIiIi1JLCVEaXZlcmdlbmNlRzYiNiMqLComIiIjIiIiXiMiIiIiIiIiIiIpLUkkYWJzRyUqcHJvdGVjdGVkRzYjLUkkcHNpRzYiNiZJInhHJSpwcm90ZWN0ZWRHSSJ5RyUqcHJvdGVjdGVkR0kiekclKnByb3RlY3RlZEdJInRHNiIiIiMiIiItSSN2X0c2IjYmSSJ4RyUqcHJvdGVjdGVkR0kieUclKnByb3RlY3RlZEdJInpHJSpwcm90ZWN0ZWRHSSJ0RzYiIiIiSSJtRzYiIiIiSScmaGJhcjtHNiIhIiIiIiJJIm1HNiIhIiJGJy9GRVE8bWFwbGVzb2Z0LmNvbTpsYWJlbChMMTY3MjcpRidGR0Y6Rk4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEkbGhzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2I1EhRictSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1FlX2wvLUkmJWRpZmZHNiI2JC1JIm5HNiI2JkkieEclKnByb3RlY3RlZEdJInlHJSpwcm90ZWN0ZWRHSSJ6RyUqcHJvdGVjdGVkR0kidEc2IkkidEc2IioqKiYjIiIiIiIjIiIiXiMiIiIiIiIiIiJJJyZoYmFyO0c2IiIiIi1JLCVEaXZlcmdlbmNlRzYiNiMqLComIiIjIiIiXiMiIiIiIiIiIiItSSJuRzYiNiZJInhHJSpwcm90ZWN0ZWRHSSJ5RyUqcHJvdGVjdGVkR0kiekclKnByb3RlY3RlZEdJInRHNiIiIiItSSN2X0c2IjYmSSJ4RyUqcHJvdGVjdGVkR0kieUclKnByb3RlY3RlZEdJInpHJSpwcm90ZWN0ZWRHSSJ0RzYiIiIiSSJtRzYiIiIiSScmaGJhcjtHNiIhIiIiIiJJIm1HNiIhIiJGJy8lK2FjdGlvbnR5cGVHUTxtYXBsZXNvZnQuY29tOmxhYmVsKEwxNjc0MClGJy8lJXZpZXdHUSZsYWJlbEYnRjovRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEifkYnRkovJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlIvJSlzdHJldGNoeUdGUi8lKnN5bW1ldHJpY0dGUi8lKGxhcmdlb3BHRlIvJS5tb3ZhYmxlbGltaXRzR0ZSLyUnYWNjZW50R0ZSLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGW28tRk02LVEiPUYnRkpGUEZTRlVGV0ZZRmVuRmduL0ZqblEsMC4yNzc3Nzc4ZW1GJy9GXW9GYm9GTC1GLDYlUSdleHBhbmRGJ0YvRjItRjY2JC1GIzYkLUYsNiVRJHJoc0YnRi9GMkY1Rko=That is,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One can still verify that, provided that there are no singularities, i.e.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GLDYtUSsmTm90RXF1YWw7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRlMtSSNtbkdGJDYkUSIwRidGL0Yv, the velocity satisfies 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, it can be written as a gradient, 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. That is, GPE admits solutions with vortices. At the center of a vortex, the field density vanishes, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzAuRidGOS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORldGOQ==This singularity warrants that the velocity circulation around a vortex is not LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkdGLw==(indeed, it is quantified, but that is beyond the scope of this worksheet).To verify that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbW9HRiQ2LVEoJm5hYmxhO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xMTExMTExZW1GJy1GLDYuUSgmdGltZXM7RidGLy9JK21zZW1hbnRpY3NHRiRRKFBoeXNpY3NGJ0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjIyMjIyMjJlbUYnL0ZFRk4tSSZtb3ZlckdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GLDYtUScmcmFycjtGJ0YvRjJGNS9GOEZZRjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGW29GPy1GLDYtUSJ+RidGL0YyRjVGN0Y5RjtGPUY/RkEvRkVGQy1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/RmpuRlxvRl1vLUkjbW5HRiQ2JFEiMEYnRi8vJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGWS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRi8=, LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== is rewritten as a function of its phase LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YvRjI= (so s is real) and amplitudeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 this value in 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Taking into account that outside the center of the vortex LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= and isolating the velocity 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The right hand side can now be conveniently rewritten as a gradient. For that purpose, compute first the inert gradient functions of 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 result can recombined as a gradient of the phase 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And from this result it follows 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The continuity equation can finally be rewritten, now carrying the information about 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, directly in terms of 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 in expanded 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[1] Gross-Pitaevskii equation (wiki)[2] Continuity equation (wiki)
[3] Bose\342\200\223Einstein condensate (wiki)[4] Bose-Einstein Condensation in Dilute Gases, C. J. Pethick and H. Smith, Second Edition, Cambridge (2008), ISBN-13: 978-0521846516.[5] Advances In Atomic Physics: An Overview, Claude Cohen-Tannoudji and David Guery-Odelin, World Scientific (2011), ISBN-10: 9812774963.[6] Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN-13: 978-0123970237.