Deriving Schrodinger's Equation w/o Boundary Conditions

[Hyperreality]

Can Schrodinger's Equation be derived without a boundary condition?

Particles according to quantum physics are only "partly localised", so does it mean that Schrodinger's equation can only be applied in a confined region of space?

Also, from what I read from my textbook, Schrodinger's Equation is applied to wave packets, because it has an "estimated" boundary of \Delta x of large magnitude. If so, how can a simple harmonic quantum oscillator exist? An ideal simple harmonic motion is represented by pure sine or cosine waves, where \Delta x = \infty.

 

[dextercioby]

Can Schrodinger's Equation be derived without a boundary condition?

Schroedinger's equation cannot be derived.It is accepted as a postulate.U mean 'solved'.In that case,it depends on the typical problem it is applied.Generally we take into account wavefunctions defined on all R^{3n}.

Particles according to quantum physics are only "partly localised", so does it mean that Schrodinger's equation can only be applied in a confined region of space?

No,Schroedinger's equation can be applied for all space,for all conditions,for all possible cases.However,not every solution to this equation describes a possible quantum state of a system.

Also, from what I read from my textbook, Schrodinger's Equation is applied to wave packets, because it has an "estimated" boundary of \Delta x of large magnitude.

Yes,it's applied to to wave packets,simply because de Broglie's plane momentum waves do not represent phyiscal states of a quantum particle as they cannot be normalized.

If so, how can a simple harmonic quantum oscillator exist?An ideal simple harmonic motion is represented by pure sine or cosine waves, where \Delta x = \infty.

It can exist,as all solution of the Schroedinger's equation can be normalized and hence describe possible quantum states.Pick one state of the QSHO and compute \Delta \hat{x} and see whether it is infinite or not...

Daniel.

 

[wizzart]

@dextercioby:

I've seen you state a few times that Schrodinger's equation cna't be derived, but needs to be accepted as a postulate or axiom. I think that would be a bit strange, since what then would bring schrodinger to formulate it? Also, I've read a derivation of the equation following from Feynman's path integral formulation of QM, that, as far as I could tell, didn't take the Schrodinger equation as a known fact. But I'm quite the newbie on path integral QM, so I could be wrong.

 

[dextercioby]

Traditional nonrelativistic QM has 3 versions/formulations:
1.Formulation with vectors and operators,due to mostly to Paul Adrien Maurice Dirac and founded mathematically by John von Neumann.
2.Formulation with operators due to John von Neumann.
3.Formulation with path-integrals due to Richard P.Feynman.

At conceptual level all these 3 formulations are equivalent.That means one implies the other and viceversa.In particular,the Schroedinger equation can be obtained from the treatment by Feynman and naturally viceversa.Actually Feynman took Schroedinger's eq.for granted and proved its complete equivalence with equations containing path-integrals.

So,in the Schroedinger picture of Dirac/traditional formulation of nonrelativistic QM,the IV-th postulate contains the Schroedinger's eq.But in the Feynman formulation,this eq.can be derived through a more tricky procedure.This formulation due to Fynman is most commonly made in the Heisenberg picture,so one has to use the equivalence between the Heisenberg picture and the Schroedinger's one to get the equation.

Anyway,the key-word is "equivalence".

Daniel.

 

[jtbell]

I've seen you [dextercioby] state a few times that Schrodinger's equation cna't be derived, but needs to be accepted as a postulate or axiom. I think that would be a bit strange, since what then would bring schrodinger to formulate it?

Schrödinger's inspiration (not a derivation in the rigorous mathematical sense) was to make an analogy between mechanics and optics. I have a PDF copy of Schrödinger's first English-language paper on the subject [Phys. Rev. 28, 1049 (1926)] following on his original German-language papers. He starts with the Hamiltonian action integral of classical mechanics:

W=\int{(T-V)dt}

and considers the family of surfaces in space on which W is constant. It turns out that the possible trajectories of a particle in that system are perpendicular (normal) to those constant-W surfaces. He comments:

The above-mentioned construction of normals dn is obviously equivalent to Huygens' principle. The orthogonal curves of our system of W-surfaces form a system of rays in our optical picture; they are possible orbits of the material points in the mechanical problem.

A bit further on:

The well-known mechanical principle due to and named after Hamilton can very easily be shown to correspond to the equally well-known optical principle of Fermat.

He discusses the relationship between geometrical (ray) optics and wave optics, and then:

Now compare with these considerations the very striking fact, of which we have today irrefutable knowledge, that ordinary mechanics is really not applicable to mechanical systems of very small, viz. of atomic dimensions.

[...] is one not greatly tempted to investigate whether the non-applicability of ordinary mechanics to micro-mechanical problems is perhaps of exactly the same kind as the non-applicability of geometrical optics to the phenonema of diffraction or interference and may, perhaps, be overcome in an exactly similar way?

[...] At any rate the equations of ordinary mechanics will be of no more use for the study of these micro-mechanical wave-phenomena than the rules of geometrical optics are for the study of diffraction phenomena. Well known methods of wave theory, somewhat generalized, lend themselves readily. The conceptions, roughly sketched in the preceding are fully justified by the success which has attended their development.

Schrödinger associates Hamilton's W-function with the phase of his \psi-function:

\psi=A(x,y,z) \sin(W/\hbar)

He assumes that this has to satisfy the usual differential wave equation, with velocity

u=E/\sqrt{2m(E-V)}

and on substituting these into the differential wave equation, out pops what we now know as the time-independent Schrödinger equation!

 

[dextercioby]

To carry on from where Jtbell left off,i'll post Schroedinger's view on the (Hydrogen) atom.

So the whole of the wave-phenomenon,though mathematically spreading throughout all space,is essentially restricted to a small sphere of a few Angstroms diameter which may be called "the atom" according to ondulatory mechanics.

However,though it has been obtained from classical phyiscs,this equation is still the basis of quantum physics and it cannot be applied to macroscopic level/classical systems.

Daniel.

 

[dextercioby]

Well,there are formulations of QM in which SE is not a postulate.Even in Dirac'ss formulation,if you postulate Heisenberg equation,you'll be able to prove SE...

Daniel.

 

[thinker]

Can Schrodinger's Equation be derived without a boundary condition?

Particles according to quantum physics are only "partly localised", so does it mean that Schrodinger's equation can only be applied in a confined region of space?

Also, from what I read from my textbook, Schrodinger's Equation is applied to wave packets, because it has an "estimated" boundary of \Delta x of large magnitude. If so, how can a simple harmonic quantum oscillator exist? An ideal simple harmonic motion is represented by pure sine or cosine waves, where \Delta x = \infty.

I have the same question.:-)

 

[ZapperZ]

Can Schrodinger's Equation be derived without a boundary condition?

Particles according to quantum physics are only "partly localised", so does it mean that Schrodinger's equation can only be applied in a confined region of space?

Also, from what I read from my textbook, Schrodinger's Equation is applied to wave packets, because it has an "estimated" boundary of \Delta x of large magnitude. If so, how can a simple harmonic quantum oscillator exist? An ideal simple harmonic motion is represented by pure sine or cosine waves, where \Delta x = \infty.

You have a considerable misunderstanding of what a "Schrodinger Equation" is, and, I suspect, a lack of knowledge on the mathematics involved. The Schrodinger equation itself is INDEPENDENT of any boundary condition. It is, after all, simply a 2nd order differential equation. It is the SOLUTIONS to the Schrodinger equation that is dependent on the boundary conditions. The solutions are the ones tailored to whatever boundaries that exist.

Does this, outright, answer your question?

Zz.

 

[marlon]

Can Schrodinger's Equation be derived without a boundary condition?

The SE is independent of any boundary condition. There is a simple way to "prove" the SE. Let's assume we have no problem with wavefunctions (why should we, right ?). Given the equation for a wavefunction (ie the exponential structure) we now that taking the second derivative with respect to position yields the momentum p squared : p² (with some constants). The first derivative with respect to time yields the energy E. Now we know that E = p²/2m so if you substitute p² and E with the derivatives of the wavefunctions, you get the SE

regards
marlon

 

[Crosson]

marlon, your derivation does nothing more than show:

\psi(\vec {r}, t) = e^i^(^\vec {p} \cdot \vec{r} + Et)

Is a solution to a particular equation, a game we could play all night (it is like asking "for what equations is 2 a solution?").

That is really a backwards derivation, and I don't see how it applies in any case other than that of a plane wave (at most, maybe you only meant one dimensional).

If you want to see the quick and dirty derivation of SE:

1) Assume there is a wave function which contains all of the observable information it is possible to know about a particle.

Cosideration: In order to find this function we must construct it by giving it properties. That is, it should be constructed as to give us information about the system when we apply a relatively simple operator.

2) Whatever the wave function is, I want to get momentum with a space derivative and energy with a time derivative, that would be simple.

Since energy and momentum are related as they are, we have the shrodinger equation! Fortunately the uniqueness of solutions to a partial differential equation gaurantees our success.

 

[dextercioby]

marlon, your derivation does nothing more than show:

\psi(\vec {r}, t) = e^{i\(\vec {p} \cdot \vec{r} + Et\)}

Is a solution to a particular equation, a game we could play all night (it is like asking "for what equations is 2 a solution?").

That is really a backwards derivation, and I don't see how it applies in any case other than that of a plane wave (at most, maybe you only meant one dimensional).

If you want to see the quick and dirty derivation of SE:

1) Assume there is a wave function which contains all of the observable information it is possible to know about a particle.

Cosideration: In order to find this function we must construct it by giving it properties. That is, it should be constructed as to give us information about the system when we apply a relatively simple operator.

2) Whatever the wave function is, I want to get momentum with a space derivative and energy with a time derivative, that would be simple.

That's both fishy and logically incorrect,because you know where you to get (the SE) and you know that the result you're expecting (SE) is correct...

Daniel.

 

[marlon]

marlon, your derivation does nothing more than show:

\psi(\vec {r}, t) = e^i^(^\vec {p} \cdot \vec{r} + Et)

I don't really think you got the point. However, since i don't want to play with you all night long, let us just drop it...ok ?

marlon

 

[EL]

well, it's not the first time this is discussed here...

 

[dextercioby]

True,these subjects in Quantum Physics forum are kinda circular.The EPR gedankenexperiment is still the most debated.

Daniel.

 

[marlon]

True,these subjects in Quantum Physics forum are kinda circular.The EPR gedankenexperiment is still the most debated.

Daniel.

That and Afshar also

marlon

 

[pmb_phy]

Can Schrodinger's Equation be derived without a boundary condition?

Not that I know of. Its only possible by way of using a different set of postulate and then the Schroedinger's equation simply fall out of it.

Pete

 

[marlon]

Not that I know of. Its only possible by way of using a different set of postulate and then the Schroedinger's equation simply fall out of it.

Pete

No, once again: the Schrodinger equantion (SE) is independent of boundary conditions. The SE is a mere manifestation of E = p²/2m, translated into derivative operators working on a wavefunction. The solutions (ie the wavefunctions) are dependent on boundary conditions in order to assure their normalizability (so they are "physical" if you will) and their continuity at boundaries (like in the case of the potential-well)

marlon

 

[dextercioby]

Marlon,SE refers to the abstract equation which is postulated.You've been referring to (not only in this thread) is the simple equation found by Erwin Schrödinger in 1926.Dirac axiomatized and abstractized everything...

Daniel.

 

[marlon]

Marlon,SE refers to the abstract equation which is postulated.You've been referring to (not only in this thread) is the simple equation found by Erwin Schrödinger in 1926.Dirac axiomatized and abstractized everything...

Daniel.

I think this is a lot of words to say nothing real though. If someone talks about the SE, i think there is a clear consensus on what he/she is referring to...Let's keep it easy and simple...You know that ignorance likes to hide behind a posh-sounding vocabularium...

marlon

 

[masudr]

Sorry marlon, but must take sides with Daniel here. the schrodinger equation is specifically that the hamiltonian generates infinitesimal translations in time (for the state vector). That is quantum mechanics. Speaking of p^2/2m and suchlike applies to a restricted set of systems.

 

[dextercioby]

Yes,ignorance toward people,not ideas...

However,you're not affected.I don't ignore you. Not yet

Daniel.

 

[reilly]

It's really quite simple. As jtbell elegantly points out, the SE can be made very plausible by using Hamiltonian theory. Recall that Hamilton-Jacobi theory played a huge role in the development of the Old QM, and, in fact, advanced mechanics was very much a standard part of physicist's working knowledge in the early days of QM.

Derived? Are Newton's Laws derived? Are Maxwell's Eq. derived?

Mathematicians talk about well set problems -- i.e. ones that have a solution. To make the SE work, as is also the case with differential and partial differential equations, you need boundary conditions. That's all she wrote.

Regards,
Reilly Atkinson

 

[Hans de Vries]

Can Schrodinger's Equation be derived ...

Schrodinger's Equation can be derived entirely from the observation that
E = hf and by postulating the use of complex frequencies e^-iEt.


1) Use Special Relativity plus the rest mass energy to expand time to 4D
space-time and energy to 4D momentum.

2) Approximate the relativistic kinetic energy by its first term only.

3) Use the gauge invariance for an arbitrary phase shift to throw out the
rest mass energy and to ignore that potential energy is defined up to
an arbitrary constant.


Never forget that throwing out the rest mass makes Schroedinger's Equation
incompatible with E₀ = hf₀ = m₀c² . Think ψ --> ψ.exp(-i ω₀t) for the
compatibility.


Regards, Hans

 

[Crosson]

Derived? Are Newton's Laws derived? Are Maxwell's Eq. derived?

I suggest that you read Gravitation by Misner, Thorne and Wheeler. This is the ultimate text for anyone who loves physics. In chapter 5 they show how conservation of energy (i.e. vanishing 4-divergence of the stress energy tensor) leads to 1)Maxwells equations 2)The Lorentz force law 3)Equation of motion for an ideal fluid and 4)conservation of angular momentum.

Of course, how do we know energy is conserved? Well, in chapter 16 it turns out that this is a consequence of a much more general, purely geometric theorem "The boundary of a boundary is zero". Great stuff!

 

[reilly]

Crosson --I admit that I'm not up on GR and the book Gravitation as much as I should be. Now, do remember that conservation of energy's validity is a matter of experiment. Theory's job is to explain why that is so. The conservation of energy is a fact of nature, any "derivation" of conservation of energy, whether in Newtonian form, or in 4-D form starts with the knowledge that energy is conserved, and seeks to understand the specific circumstances in which energy is conserved or not conserved. The notion of electric and magnetic fields also is based on experiment. But, Maxwell's deduction of displacement current can, I think, properly be termed a derivation. How would you know whether Coloumb's law, or Poisson's eq. are correct without experiment?

Dirac's invention of his equation can rightfully be called a derivation, one of the most brilliant in all of history. Most symmetry arguments are ultimately based on empirical evidence, and are not derivations. (The theory of angular momentum coupling is indeed based on mathematical arguments and derivations, which give the 3-,6- and 9- j symbols and all that stuff. )And so on.

In addition to MTW, anyone who "loves physics" should read Dirac's Quantum Mechanics, Lanczos The Variational Principles of Mechanics, and, perhaps, Landau and Lif****z's Classical Theory of Fields.(The stars are terminally stupid. Perhaps someone can remove the censorship of a brilliant physicist's name)

(I will admit that I was trained as a dirt-farmer theoretician, and have a highly emprical approach to physics. I also know that there are other perspectives that work well for those who hold them.)

Regards,
Reilly Atkinson

 

[marlon]

Sorry marlon, but must take sides with Daniel here.

I wish you the best of luck...

the schrodinger equation is specifically that the hamiltonian generates infinitesimal translations in time (for the state vector).

Untrue. That is the time dependent SE. How about the time independent SE ?
You are just talking about time evolution but that is not what we are talking about.
Let me ask you this question : why is time evolution unitary?

I think this proves you took the wrong side

That is quantum mechanics.

Really ?

Speaking of p^2/2m and suchlike applies to a restricted set of systems.

Oh really, then given me an example of such a "restricted" system...

I am curious

regards
marlon

 

[dextercioby]

Time independent equation is a particular case to the time dependent one.It assumes time-independent Hamiltonian------->stationary states---------->the possibility of factoring out the time dependence of the state vector...

Daniel.

P.S.What's the Hamiltonian for a charged particle in an EM field...?

 

[marlon]

P.S.What's the Hamiltonian for a charged particle in an EM field...?

Don't you know that ?

marlon

it is in my journal, check it out...

 

[dextercioby]

Apparently u don't.You asked Masud why not \frac{p^{2}}{2m} as a kinetic part of a the Hamiltonian.I've given you the example in which the kinetic part is more than that & incidentally it is time dependent.

Daniel.

 

[marlon]

Apparently u don't.You asked Masud why not \frac{p}^{2}{2m} as a kinetic part of a the Hamiltonian.I've given you the example in which the kinetic part is more than that & incidentally it is time dependent.

Daniel.

This is irrelevant. The p²/2m-part is still there...

marlon

 

[reilly]

Some history

I wish you the best of luck...



Untrue. That is the time dependent SE. How about the time independent SE ?
You are just talking about time evolution but that is not what we are talking about.
**********************************
Once again, a little history is in order. The idea of the Hamiltonian generating displacements in time is old, like from the 19th century -- Jacobi's transformation theory, contact transformations and the like. The idea that time dependent and time independent approaches were inextricably connected was already time-honored by the 1920s when Heisenberg and Schrodinger invented modern QM. Further, the theory of wave-like motion, elastic strings membranes, even E&M in it's Lagrangian formulation contributed greatly to the connections between time dependent and independent approaches -- like through eigenstate expansions. (Lanczos, Goldstein in their books on mechanics discuss these issues. Virtually all of physics is about time evolution, and stationary states are exceptions - except, -perhaps for isolated atoms and nuclei. )

(There's a nice discussion about this in Pais' biography of Bohr)

So, indeed, we are talking about time evolution.
*****************************************

Let me ask you this question : why is time evolution unitary?
*******************************
Any QM book will explain that QM time evolution must be unitary to preserve probability, which requires the QM Hamiltonian operator to be Hermitian. (See, in particular, Dirac's QM book.)

Again, jtbell got it right, elegantly so, and is completely consistent with the thinking of physicists in the late 1920s and early 1930. The SE, time dependent or not, cannot be derived, but it can be made very plausible.

Regards,
Reilly Atkinson

 

[masudr]

Marlon, there are indeed two kinds of SE, time-dependent and time-independent. The second one is representative of the fact that eigenstates of the Hamiltonian are energy eigenstates; the energy analogue of the fact that eigenstates of the position operator are position eigenstates (!) which is obvious. What the time-independent equation says is that the Hamiltonian is the operator associated with energy. And it is only valid for energy eigenstates.

The time-dependent equation is making a much more serious claim. It is describing how all state vectors evolve, not just energy eigenstates. As Daniel points out, p^2/2m is not the general case. Saying that is the case is like saying the Lagrangian of all systems is given by L = T - V. This is true for most cases, but not for all. The same applies with saying the p^2/2m form of the SE is valid for all cases -- it is true for most but not all.

The Schrodinger equation cannot be "derived" -- it's validity can be made plausible, but some things still have to be assumed. In a model of the universe, some things have to be assumed, and the SE is one of them for the quantum mechanical model of the universe.

Someone has tried to explain Newton's laws as a consequence of some other principle from General Relativity. Well, let's first be clear on this -- Newton's Laws are merely stating the conservation of linear momentum in any inertial frame. Considering that GR builds on this fact and generalises it, it is no surprise that Newton's Laws drop out of some higher principle from GR. But of course, that is a different model of the universe. There are still certain things that have to be assumed in GR. And any model will have to have at least one assumption that must be accepted.

 

[marlon]

As Daniel points out, p^2/2m is not the general case.

Yes i know that and i never said that p²/2m is the most general case. I don't think you got my point. I just wanted to illustrate how you can give an intuitive "proof" as to why the SE has that specific form.

This remark of yours is something everybody knows and i have never denied this.

The Schrodinger equation cannot be "derived" -- it's validity can be made plausible, but some things still have to be assumed.

In my opinion it can (see my previous posts). The example i gave using only p²/2m clearly illustrates my point. The fact that this is not the general case does not make the content of my point wrong. Interactions are just added using potentials, so i really don't see what the problem is.

The most basic postulate is the wavefunction...

Also the time evolution can easily be proved in a more 'mathematical' manner.

Now let us not restart this discussion again. It will lead to nothing and let us just conclude that we have a difference in opinion.

marlon

 

[marlon]

Let me ask you this question : why is time evolution unitary?
*******************************
Any QM book will explain that QM time evolution must be unitary to preserve probability, which requires the QM Hamiltonian operator to be Hermitian. (See, in particular, Dirac's QM book.)

Correct

marlon

 

[masudr]

Also the time evolution can easily be proved in a more 'mathematical' manner.

I'm interested, please show me.

Now let us not restart this discussion again. It will lead to nothing and let us just conclude that we have a difference in opinion.

Yes, I agree.

The most basic postulate is the wavefunction...

Note, this is more or less unrelated to this discussion, but (a) by wavefunction, I assume you mean state vector, and (b) the three postulates: the state is represented by a state vector in a Hilbert space, all observables are represented by Hermitian operator (or at least normal operators), and the time-evolution of the state vector are all equally important in my opinion.

The reason is that these three are important is because of the link to Hamiltonian mechanics where, the postulates are that the state is represented by a point in a symplectic phase space, the operators are all real-valued function on that space, and the Hamiltonian gives the time-evolution of the point in phase space.

End of discussion.

 

[dextercioby]

Did i miss anything...? Apparently 5 posts.

Daniel.

 

[marlon]

I'm interested, please show me.

Sighs, ok here we go again.

Like i showed it is proven that the time derivative of the wavefunction yields the Hamiltonian multiplied by this wavefunction. The Hamiltonian is just some letter H equal to p²/2m + V...This has been well established in my first and second post.

Now time evolution (being unitary ofcourse for the well known reasons) is easy to derive when knowing that the evolution is expressed by a unitary opertaor U that works on a "given" wavefunction. Substitute this in the SE and solve the corresponding integral equation.

Besides, the unitarity of this U is closely related to the hermiticity of H. you can display this connection by studying the change in this U induced after an arbitrary small time delta t. This proves that H is therefore the generator of an infinitesimal unitary transformation in time expressed by U(t+delta t, t)...and so on and on...

read Bransden and Joachain page 232 and further...

End of discussion.

Please, i insist...
Can i take your word on that ?

Well, we shall see

marlon

 

[dextercioby]

All 5 postulates are of equal importance...One missing,and the description would be incomplete.

Daniel.

 

[Hans de Vries]

Schrödinger's Equation derived from E = hf

The SE, time dependent or not, cannot be derived, but it can be made very plausible.

The Schrodinger equation cannot be "derived" -- it's validity can be made plausible,
but some things still have to be assumed.

Of course we can derive Schrödinger's equation. It does not provide
anything new which can't be derived from more basic physics.




==============================================
Schrödinger's Equation derived from E = hf
==============================================


Step 1:
We get the complex frequencies e^-iEt/ħ by interpreting the Fourier Transform
over time as the energy spectrum. This gives us the derivatives:

from: Ψ = e^-i2πft = e^-iωt = e^-iEt/ħ and: ∂Ψ/∂t = -iωe^-iωt

it follows for the derivatives:

EΨ = iħ ∂Ψ/∂t
E²Ψ = -ħ² ∂²Ψ/∂t²


Step 2:
All we need to get the complementary p = h/λ is using Special Relativity.
The (relativistic) deBroglie wave-length is caused purely by non-simultaneity:
The time-shift we see if we look at the particle from a reference frame in
which it is not at rest:

Ψ(x₁,t₁) = e^-iEt₁
Ψ(x₂,t₂) = e^-iEt₂

Only in the rest frame we have t₁=t₂ and the phase of Ψ(x,t) is constant in x.
After applying the Lorentz Transform ct' = γ (-βx + ct) we get x mixed in:

Ψ(x) = e^i2πx/λ = e^ipx/ħ

It follows for the derivatives:

pΨ = -iħ ∂Ψ/∂x
p²Ψ = -ħ² ∂²Ψ/∂x²



Step 3:
Special Relativity would lead us straight to the Klein Gordon equation with:
E² = p²c² + m²c⁴, where m is the rest-mass. We need the non-relativistic
version however and we also need to include the potential energy:

E² = p²c² + (U + mc²)²

We approximate the energy via:

E = (U + mc²) √( 1 + p²c²/(U+mc²)² )
E ≈ (U + mc²) √( 1 + p²/(m²c²) )
E ≈ (U + mc²) (1 + p²/(2m²c²) )
E ≈ U + mc² + p²/(2m)

This relation holds for all space-time points of the wave function Ψ(x,t):

EΨ = (U + mc² + p²/(2m) )Ψ

This now gives us the complete non relativistic equation:

iħ ∂Ψ/∂t = (U + mc²)Ψ - ħ²/(2m) )∂²Ψ/∂x²


Step 4:
We're now only a single step away from the Schrödinger's equation.
We define Schrödinger's wave function as:

Ψ_s = e^{i(U₀ + mc2)t/ħ} Ψ_c

Where U₀ is an arbitrary potential energy constant and Ψ_c is the correct
version. This is a bit ugly but it turns out that it doesn't matter for the
probabilities coming out of the time-dependent version, since:

Ψ_s^*Ψ_s = e^{-i(U₀ + mc2)t/ħ} Ψ_c^* . e^{i(U₀ + mc2)t/ħ} Ψ_c = Ψ_c^*Ψ_c

And of course it doesn't the matter for the time independent version.
So after removing the rest mass energy term (which is what this accounts
to) we end up with the actual Schrödinger's equation:

iħ ∂Ψ/∂t = UΨ - ħ²/(2m) )∂²Ψ/∂x²

based only on the observation that E = hf



Regards, Hans

 

[dextercioby]

Of course we can derive Schrödinger's equation. It does not provide
anything new which can't be derived from more basic physics.




==============================================
Schrödinger's Equation derived from E = hf
==============================================


Step 1:
We get the complex frequencies e^-iEt/ħ by interpreting the Fourier Transform
over time as the energy spectrum. This gives us the derivatives:

from: Ψ = e^-i2πft = e^-iωt = e^-iEt/ħ and: ∂Ψ/∂t = -iωe^-iωt

it follows for the derivatives:

EΨ = iħ ∂Ψ/∂t
E²Ψ = -ħ² ∂²Ψ/∂t²


Step 2:
All we need to get the complementary p = h/λ is using Special Relativity.
The (relativistic) deBroglie wave-length is caused purely by non-simultaneity:
The time-shift we see if we look at the particle from a reference frame in
which it is not at rest:

Ψ(x₁,t₁) = e^-iEt₁
Ψ(x₂,t₂) = e^-iEt₂

This is incorrect.The energy is not a Lorentz scalar...

Only in the rest frame we have t₁=t₂ and the phase of Ψ(x,t) is constant in x.
After applying the Lorentz Transform ct' = γ (-βx + ct) we get x mixed in:

Ψ(x) = e^i2πx/λ = e^ipx/ħ

It follows for the derivatives:

pΨ = -iħ ∂Ψ/∂x
p²Ψ = -ħ² ∂²Ψ/∂x²



Step 3:
Special Relativity would lead us straight to the Klein Gordon equation with:
E² = p²c² + m²c⁴, where m is the rest-mass. We need the non-relativistic
version however and we also need to include the potential energy:

E² = p²c² + (U + mc²)²

We approximate the energy via:

E = (U + mc²) √( 1 + p²c²/(U+mc²)² )
E ≈ (U + mc²) √( 1 + p²/(m²c²) )
E ≈ (U + mc²) (1 + p²/(2m²c²) )
E ≈ U + mc² + p²/(2m)

This relation holds for all space-time points of the wave function Ψ(x,t):

EΨ = (U + mc² + p²/(2m) )Ψ

This now gives us the complete non relativistic equation:

iħ ∂Ψ/∂t = (U + mc²)Ψ - ħ²/(2m) )∂²Ψ/∂x²


Step 4:
We're now only a single step away from the Schrödinger's equation.
We define Schrödinger's wave function as:

Ψ_s = e^{i(U₀ + mc2)t/ħ} Ψ_c

Where U₀ is an arbitrary potential energy constant and Ψ_c is the correct
version. This is a bit ugly but it turns out that it doesn't matter for the
probabilities coming out of the time-dependent version, since:

Ψ_s^*Ψ_s = e^{-i(U₀ + mc2)t/ħ} Ψ_c^* . e^{i(U₀ + mc2)t/ħ} Ψ_c = Ψ_c^*Ψ_c

And of course it doesn't the matter for the time independent version.
So after removing the rest mass energy term (which is what this accounts
to) we end up with the actual Schrödinger's equation:

iħ ∂Ψ/∂t = UΨ - ħ²/(2m) )∂²Ψ/∂x²

based only on the observation that E = hf



Regards, Hans

Daniel.

 

[Hans de Vries]

This is incorrect.The energy is not a Lorentz scalar...
Daniel.

?

p = h/λ follows from E = hf via the Lorentz Transform.
That's as correct as it can be. What I want to illustrate
here is that a pure harmonic:

e^-iEt/ħ in it's rest frame.

transforms to

e^{ip'x'/ħ-iE't'/ħ}

under Lorentz transform to a frame in which it moves.
There's no need to separately postulate p = h/λ.



Regards, Hans

 

[dextercioby]

That:
Ψ(x1,t1) = e^(-iEt1)
Ψ(x2,t2) = e^(-iEt2)

is incorrect,why don't u get it...?

Daniel.

 

[Hans de Vries]

That:
Ψ(x1,t1) = e^(-iEt1)
Ψ(x2,t2) = e^(-iEt2)

is incorrect,why don't u get it...?

Daniel.

What do you want it to be then? The deBroglie wave-length is a relativistic
effect which occurs at cm/second or mm/second speeds. Maybe you want
the relativistically corrected value for E? Doesn't make a lot of difference here.

It's purely to illustrate that the time shift produces a different phase shift in
e^-iE₀t/ħ at different x locations which itself then manifest as the deBroglie
wave length. E₀ is the rest mast energy.

Next time I'll give the formal prove which is not that hard to do.
Or you can do it yourself ??


Regards, Hans.

 

[dextercioby]

Why are there 2 functions...?Psi with subscripts "1" & Psi with subscripts "2".Is it the same wavefunction,but in two different reference frames...?

Daniel.

 

[marlon]

Great post Hans,
Finally somebody who really understands QM. Your derivation is indeed the longer version of what i have been trying to point out.

I have one remark though. I am not following step 2 with those :
Ψ(x1,t1) = e-iEt1
Ψ(x2,t2) = e-iEt2

Do we really need them ?

regards
marlon, ps : great work

 

[Hans de Vries]

I have one remark though. I am not following step 2 with those :
Ψ(x1,t1) = e-iEt1
Ψ(x2,t2) = e-iEt2

Do we really need them ?

regards
marlon, ps : great work

They're not really needed no. I replaced them with some text and made
a Journal entry from the derivation:

================================
Schrödinger's Equation derived from E = hf
================================

https://www.physicsforums.com/journal.php?s=&journalid=13459&action=view


It now also includes the derival of p = h/λ from E = hf via the Lorentz
Transform. (Which is just a "one-liner" really)


Regards, Hans

 

[masudr]

Finally somebody who really understands QM.

Why are you making this personal? I don't understand. I find that offensive, to be honest. I'll point it out, and then let it rest. I just don't see why this discussion should ever degenerate to personal tit-for-tat style arguments.

 

[marlon]

Why are you making this personal? I don't understand. I find that offensive, to be honest. I'll point it out, and then let it rest. I just don't see why this discussion should ever degenerate to personal tit-for-tat style arguments.

Why would you even feel attacked ?

Hans finally gave a nice discription of the actual "birth" of the SE in QM. This is also something i have been pointing out here many times (also in previous threads). Saying the SE cannot be derived implies it just "fell out of the sky"
and this is what i mean by not understanding QM. IT does not fall out of the sky in terms of posh-sounding postulates that cannot be discussed.

marlon

 

[reilly]

How's that again?

No, once again: the Schrodinger equantion (SE) is independent of boundary conditions. The SE is a mere manifestation of E = p²/2m, translated into derivative operators working on a wavefunction.
*************************************
What? You might consider explaining this notion. Yours is quite an unusual approach.
**************************************************
The solutions (ie the wavefunctions) are dependent on boundary conditions in order to assure their normalizability (so they are "physical" if you will) and their continuity at boundaries (like in the case of the potential-well)

**************

Whether boundary conditions are part of the SE is a somewhat of a problematical matter. If, for example, you use a Laplace Transform, then the boundary conditions become part of the transformed equation, as would be the case with a "one-sided Fourier transform" But, a regular FT does not explicitely contain boundary conditions. But, there are integrability conditions with the FT, which can be tantamount to Sommerfeld's Radiation Condition, a standard requirement for solutions as their spatial argument goes to infinity.

Regards,
Reilly Atkinson