Now what precisely is that this equation wherein does it absolutely come from and what vital
records does this equation offer us with those are a number of the questions
we are going to talk about on this lecture, so let's start through recalling a very
vital precept in quantum mechanics, so quantum mechanics essentially incorporates
the wave-particle duality of count number that may be a machine can act as a wave or it
can act as a particle now in quantum mechanics we describe any machine consisting of
an electron through the use of an amount referred to as the wave feature now while we deal with
our structures consisting of our electron as a wave if so the wave feature represents
the displacement or amplitude of the wave produced through that electron at the
different hand if we deal with our machine yet again, for instance our electron as a
particle, then in this example the wave feature lets in us to calculate the
opportunity of locating that particle, that electron at a few second in time and
a few areas in area extra specially to calculate the opportunity, we take
the and of absolutely the fee of the wave feature that's through the manner given
through the Greek letter SAI now what precisely does this inform us, nicely essentially in
order in an effort to describe any machine consisting of our electron in quantum
mechanics we ought to be capable of decide the wave feature numerically and that is
precisely wherein the Schrödinger equation absolutely comes into play, the shore
equation is largely a very crucial equation that we use to remedy for the
wave feature portions Li, so essentially the wave feature is a method to these
Equation, which itself is a differential equation now
the query is wherein precisely does the Schrödinger equation absolutely come from
are we able to absolutely derive the Schrödinger equation from a few underlying precepts
nicely with inside the identical manner that Sir Isaac Newton invented the second one regulation of movement so essentially Isaac Newton invented F equals M instances to explain what takes place to an item while an internet pressure acts on it after which that equation become essentially showed experimentally the display access equation become additionally invented and become examined and showed experimentally so because of this there may be no manner to absolutely derive the Schrödinger equation with inside the identical actual manner there is no manner to absolutely derive the equation F equals M instances
a those equations are essentially equations that may be effectively showed experimentally and can't absolutely be derived in any manner now the query nonetheless stays what does the Schrödinger equation decide the shape that the equation takes we are able to use the conservation of electricity
So that is the concept that we are going to use to decide what the Schrödinger
equation is now on this specific lecture, we are able to additionally anticipate that the wave
feature that we're coping with does now no longer absolutely depend upon time and the wave
feature most effective relies upon at the spatial role of our machine mendacity, the x-axis so
sigh, the wave features most effective relies upon on X and is impartial of the time so that
essentially way for the reason that display damage equation has an answer, this is the same to
the wave feature and the wave feature is time impartial which means the display
linear equation that we are going to decide on this lecture, is likewise time
impartial now a 2nd shape, an extra complex shape of the Schrödinger equation
that we are going to deal with in a destiny lecture is the time structured showed
into your equation we are the most effective going to attention at the time impartial short
damage equation, so let's start our dedication of this equation through searching at
the subsequent diagram let's think that a particle is transferring, which means our
particle is in reality an unfastened particle, so for instance we've an electron this is
transferring alongside the x-axis in an advantageous path, no forces act on that particle
so, which means that the electricity of a particle is steady and that means that
the momentum of particle is likewise regular now we recognize that Louis de Broglie essentially advised us that our momentum P of our particle is identical to H divided through lambda now if we take our equation and clear up for lambda the wavelength of the wave produced through the particle this is identical to H divided through P, so we see that due to the fact that H is a regular and P is likewise a regular due to the fact no forces are assumed to behave at the particle then for this unique case the wavelength is likewise regular with the intention to essentially describe the wave produced through a particle consisting of this electron shifting alongside the x-axis we would assume that the wave feature satisfies a differential equation
This is much like the classical equation of our wave this is given through the following components so our wave feature facet with appreciate to X and T our role and time is identical to a regular accelerated through sine of K X, minus Omega T plus B every other regular accelerated through cosine of K X, minus Omega accelerated through T all over again that is the classical wave equation for mechanical and electromagnetic waves now due to the fact that we're assuming that our time is unbiased this is the lack equation is unbiased of time and best relies upon at the spatial role X alongside the x-axis we are able to set T identical to zero then this disappears, So we see that sigh our wave feature with appreciate to X is identical to a sine K X, plus B cosine K X, let's label
this equation as equation 1 so what precisely is the significance of equation 1 what does it a reality provide us nicely this equation offers us the equation of the wave feature for an unfastened particle a particle that doesn't in reality experience any pressure and this equation is critical as it offers us the answer to the Schrödinger equation so essentially regardless of the tide unbiased Schrödinger equation in reality looks as if that equation has to have an answer that satisfies equation 1, and this is the significance of this equation now what precisely is K nicely K is a regular, and it is identical to two pi divided through lambda we recognize that from classical physics now lambda from this equation is H divided through P, so we move from this equation to this equation through changing lambda with H divided through P now we are able to take the 2 through of the 2 pi, and produce it to the lowest, and we see that K is identical to P divided through H divided through 2 pi, so this denominator is identical to H bar
what in which H bar is a regular, and is identical to at least one factor 0 5 instances 10 to
poor 34 joules accelerated through seconds now, so K is identical to P divided through H
bar so K is a regular and K is given through those portions so now let's move
Directly to step, so in step we are nevertheless searching at this diagram. So now let's
think we need to use the conservation of electricity so due to the fact that this particle is
shifting alongside the x-axis with inside the high-quality route with a speed this is
regular given through V and a mass M then the whole electricity in their particle our
machine e is identical to the sum of the kinetic electricity of the particle and the
capacity electricity so E is identical to K Plus u now on this lecture we are going to
attention on non-relativistic kinetic electricity so which means the non-relativistic
kinetic electricity K is identical to at least one-1/2 of M instances V squared now what precisely is V nicely remember that our momentum P is identical to M instances V, and that means that our V is identical to P divided through M, so we are able to update our V with P divided through M, and we get this end result so II the whole electricity
of our particle is identical to P squared divided through 2 m plus U the capacity electricity
in which that is our non-relativistic kinetic electricity now from this equation we recognize that K is identical to P divided through h-bar now we are able to rearrange this equation, and we see that P is identical to K accelerated through H bar, so now we update our P
with K accelerated through H bar and this offers us equation 2, so we label this as
equation 2, so we noticed in step one which equation 1 offers us the overall shape of
the answer the wave feature that's the answer of the Schrödinger equation and in step 2 what this equation in reality tells us is it tells us that determined this equation after which to truly affirm and take a look at whether or now no longer this equation truly works we ought to behavior experiments and notice whether or now no longer this equation truly suits
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