# Does schrodinger equation give complete description of electron?

1. Mar 27, 2012

is schrodinger equation capable of explaining all the properties of electron dynamics or are there any anomaly..?

2. Mar 27, 2012

### Staff: Mentor

No. The Schrödinger equation does not include the electron's "spin", also it is non-relativistic. To fix both problems, use the Dirac equation instead.

3. Mar 27, 2012

### dextercioby

Depends on what you mean by Schrödinger's equation. In 1926 he published 2 equations for the electron, one of them took into account the theory of special relativity, while both of them didn't account for the electron's spin angular momentum. Dirac in 1928 wrote a better equation.

4. Mar 27, 2012

### Demystifier

For some reason, that relativistic Schrodinger equation is today called Klein-Gordon equation.

5. Mar 27, 2012

### M Quack

That is misleading. The KG equation is relativistic, but describes a spin-zero particle. The electron has spin 1/2 and the relativistic equation to best describe it is the Dirac equation.

6. Mar 27, 2012

### tom.stoer

And even the Dirac equation has been re-interpreted as an operator equation in quantum field theory (quantum electrodynamics) instead of relativistic quantum mechanics

7. Mar 27, 2012

### Bob S

The relativistic Schrodinger (KG) equation describes the pionic atom very accurately, and measurements of pionic x rays in calcium and titanium were used to determine the mass of the pion. See Robert E. Shafer Phys. Rev. 163, 1451 (1967).

8. Mar 28, 2012

### Demystifier

To summarize all this in a confusing way:

The relativistic Schrodinger equation of electron is not really called Schrodinger equation (but Klein-Gordon equation), does not really describe electron (because it does not include spin) and is not really consistent (because it does not conserve probability). The consistent relativistic electron equation is Dirac equation, which includes spin and conserves probability. But even this equation is not fully correct, because one needs to second quantize it, and in turns out that conservation of probability is no longer conservation of probability but conservation of charge. But it also turns out that the Klein-Gordon equation also needs to be second quantized, so it turns out that non-conservation of probability becomes irrelevant for a similar reason as conservation of probability for the Dirac equation. So the Dirac equation is not important because it conserves probability, but because it derives spin from linearization of the Klein-Gordon equation. But to derive spin you don't really need to linearize Klein-Gordon equation, because you can also get spin from linearization of the non-relativistic Schrodinger equation, which is called simply Schrodinger equation, and by the way, also needs to be second quantized. But derivation of spin from linearization is not really a true derivation of spin, because the true derivation of spin comes from irreducible representations of the rotation group. But actually not of the proper rotation group SO(3), but of its simply connected covering group SU(2).

If you think that's it, you are wrong. Actually it is misleading to say that all these equations above need to be second quantized, because there is only one quantization, but applied to different degrees of freedom. So what we called second quantization, was really second quantization of particles, and is actually first quantization of fields. So fundamental objects are fields, not particles. Or maybe not, because we measure particles, not fields. But not always, because sometimes we really measure fields, but only bosonic fields. Fermionic fields, on the other hand, cannot be measured even in principle.

Further complication comes from use of quantum field theory in condensed-matter physics, where it suggests that fields are not fundamental at all, not even bosonic ones, because the field description is appropriate only at large distances. Particles are more fundamental in condensed-matter physics, so quantum field theory is better called second quantization there. But the second-quantized particles in condensed-matter physics are actually pseudo-particles (e.g., phonons), not the fundamental particles. The fundamental particles in condensed-matter physics are atoms and molecules, for which we know that they are not fundamental at all, because they consist of fundamental quarks and electrons, which are fundamentally described by another quantum field theory, which, by being fermionic field theory, describes fermionic fields which cannot be measured even in principle.

Further complication arises if you include the insight from string theory, but are you sure that you want to see that too?

Last edited: Mar 28, 2012
9. Mar 28, 2012

### M Quack

Now the confusion is complete :-)

Without going into the details, this is how I see the situation before most of the complications arise:

Classically, the energy of a particle can be written as $E=\frac{p^2}{2m}+ V$ where p is the momentum and V the potential energy.

Schrodinger replaced the momentum p by an operator $-i\hbar \vec{\nabla}$, and the energy E by the operator $i\hbar\frac{\partial}{\partial t}$. The operators act on a wave function, giving a partial differential equation that is usually called the Schrodinger equation. The wave function has only one component.

In this form the Schrodinger equation has a few shortcomings. In particular, it is not relativistic and does not describe some properties of the electron such as spin and spin-orbit coupling.

These can be added "manually" to the Schrodinger Hamiltonian. The result is sometimes called the Pauli equation. The spin operator is a complex 2x2 matrix that is a linear combination of the Pauli matrices. The wave function then has two components that describe the two spin states (e.g. spin-up and spin-down), and how they convert into each other, e.g. by precession in an external magnetic field.

The Klein-Giordon equation is derived similarly (substituting the same operators), but starting from the relativistic energy $E^2 = p^2 c^2 + m^2 c^4$ where m is the rest mass of the particle and c the speed of light. The equation is relativistic, but still does not describe the spin. It does, however, describe spin-less particle such as the pion very well. The wave function has again just one component.

The Dirac equation describes a wave function with 4 components, 2 each for particles and antiparticles, and two each for the spin states. All operators are complex 4x4 matrices. The Pauli equation can be derived as non-relativistic limit of the Dirac equation. This operation is called the Foldy-Wouthysen transformation. It shows that spin and spin-orbit coupling are relativistic effects.

Neither of these equations describes the creation and annihilation of particles, for example the creation of a photon upon the transition from an excited to the ground state. For this one needs to introduce what is called the second quantization, but that is another can of worms, and a pretty big one at that.

Wikipedia has fairly nice entries on the Klein-Gordon and Dirac equations. Check them out.

Last edited: Mar 28, 2012
10. Mar 28, 2012

### Demystifier

Actually, not yet.
I have further extended it in my blog:
https://www.physicsforums.com/blog.php?b=3873 [Broken]

Last edited by a moderator: May 5, 2017
11. Mar 28, 2012

### unusualname

I love the way this thread developed rapidly from simple to quite deep and entertaining :-)

btw, spin is not fundamentally a relativistic effect, as shown by levy-leblond in 1967 and discussed a while back on this forum, http://projecteuclid.org/euclid.cmp/1103840281 (Demystifier alluded to this)

spin coupling may be considered relativistic but then to get real precision you really need to go to full qed and even that's only effective so the true story isn't known.

edit. oh and the Schrödinger equation does describe the electron, just that we fools don't know the correct hamiltonian yet ;-)

Last edited: Mar 28, 2012
12. Mar 28, 2012

### M Quack

Confusion may reach saturation well before the story is complete. It is something you can renormalize to bring back down to finite levels :-)

Last edited by a moderator: May 5, 2017
13. Mar 29, 2012

I still don't get the point concerning relativistic equation for electron..Why relativistic equation is needed to describe motion of electron..? If electron goes at comparable to light speed Its space and ofcourse time should contract :/ and at the same time it should become massive which is not the case..
PS: I m just learning quantum mechanics and learning in a hurry which may lead to SUPERPOSITION of other concepts...Hope I sound meaningful and plz correct me if I m wrong anywhere

14. Mar 29, 2012

### M Quack

In many cases the Pauli equation (Schroedinger + Spin) is perfectly sufficient. There are, however, cases where the relativistic corrections become significant. In heavy atoms like Uranium, the binding energy of the 1s electrons is nearly 100 keV, and that is not negligible compared to mc^2 = 511 keV of the electron. And that is just every-day condensed matter physics/chemistry.

What is more important is to go to the quantization of fields that allows the creation/annihilation of particles like photons to properly describe matter light interactions.

15. Mar 29, 2012

### Bob S

The best comparison I have seen between the non-relativistic (NR) and relativistic Schrodinger (Klein Gordon, KG) equations is in Schiff "Quantum Mechanics". Chapter II. Eq 6.12 uses E = p2/2m (NR), while in Chapter XII, Eq(42.2) this becomes E2 = (pc)2 + (mc2)2 (KG). The hydrogen atom KG atomic energy levels are different, and there is splitting for different orbital angular momentum l (\ell) quantum numbers (not true in NR solution). Compare Eq (16.38A) and (42.21). Both of these differ from the Dirac solution Eq.(42.27).