question about eigenfunctions

[`hýsøŕ]

in the schrodinger equation, say, or any other eigenfunction equation in quantum mechanics, what does it mean when the wavefunction is not an eigenfunction of the hamiltonian operator (or the operator in question)? does that mean that you cannot measure the energy or that its variable or what? :S confuses me

[swansont]

If the operator in question doesn't commute with the Hamiltonian, it basically means that the measurement changes the energy — you won't know what the energy is after the measurement, and if two measurements are done in the opposite order you will get a different answer.

 

One scenario is that the wavefunction is a superposition of energy states, so when you measure the energy you'll get an energy eigenstate/eigenvalue, but measuring the other variable puts it back in an energy superposition.

[studiot]

The Schrodinger equation can be writtern in terms of the Hamilton operator H(_) and the total energy operator E(_).

 

[math]H(\psi ) = E(\psi )[/math]

 

Now this is a partial differential equation that has functions as solutions.

 

So any solution, [math]\psi [/math], is a function of the space variables in the original equation.

As with any differential equation we pick out desired solutions by imposing boundary conditions.

 

If these boundary conditions lead to a set of solutions only for discrete, separated values of E, that is they constrain E only to certain specific values (remember E is a scalar) then the solutions are called 'eignefunctions' and the values the 'eigenvalues'.

 

Some boundary conditions, however, allow solutions, [math]\psi [/math], that can take any value of E (perhaps within certain min and max limitis).

In this case there is a continuous spectrum and the solutions are not eigenfunctions.

 

So if we take the so called Morse energy curve for a quantum oscillator (from Wikipedia)

 

 

We see two regions in the graph.

The Schrodinger equation applies to both parts.

The lower part shows discrete separate energy levels getting closer together as we rise up the vertical axis, until there is a second region labelled 'dissociation energy' and a continuous emitted spectrum.

 

The energy levels in the lower region correspond to eigensolutions.

 

The continuous range of allowable energies in the upper region correspond to non eigensolutions.

Edited May 16, 2014 by studiot

[`hýsøŕ]

If the operator in question doesn't commute with the Hamiltonian, it basically means that the measurement changes the energy — you won't know what the energy is after the measurement, and if two measurements are done in the opposite order you will get a different answer.

 

One scenario is that the wavefunction is a superposition of energy states, so when you measure the energy you'll get an energy eigenstate/eigenvalue, but measuring the other variable puts it back in an energy superposition.

 

 

i see, thanks. this is very unlike anything in classical mechanics, though i guess i was warned beforehand

 

The Schrodinger equation can be writtern in terms of the Hamilton operator H(_) and the total energy operator E(_).

 

[math]H(\psi ) = E(\psi )[/math]

 

 

Now this is a partial differential equation that has functions as solutions.

 

So any solution, [math]\psi [/math], is a function of the space variables in the original equation.

 

As with any differential equation we pick out desired solutions by imposing boundary conditions.

 

If these boundary conditions lead to a set of solutions only for discrete, separated values of E, that is they constrain E only to certain specific values (remember E is a scalar) then the solutions are called 'eignefunctions' and the values the 'eigenvalues'.

 

Some boundary conditions, however, allow solutions, [math]\psi [/math], that can take any value of E (perhaps within certain min and max limitis).

In this case there is a continuous spectrum and the solutions are not eigenfunctions.

 

So if we take the so called Morse energy curve for a quantum oscillator (from Wikipedia)

 

400px-Morse-potential.png

 

We see two regions in the graph.

The Schrodinger equation applies to both parts.

The lower part shows discrete separate energy levels getting closer together as we rise up the vertical axis, until there is a second region labelled 'dissociation energy' and a continuous emitted spectrum.

 

The energy levels in the lower region correspond to eigensolutions.

 

The continuous range of allowable energies in the upper region correspond to non eigensolutions.

 

 

so the eigenfunctions will give a set of discrete eigenvalues and if instead you get a continuous range of values for E that means that isn't an eigenfunction, because if it was a continuous range, then H≠E right?

Edited May 16, 2014 by `hýsøŕ

[studiot]

Yes that's the size of it, but remember that the solutions not only satisfy the equation but also the boundary conditions.

 

There are many more solutions that satisfy the equation, but not the boundary conditions.

Edited May 16, 2014 by studiot

[Enthalpy]

The proper functions of the Hamiltonian are the ones with well-defined energy. If you measure their energy, you get always this one value.

 

The wave can differ from the Hamiltonian's proper ones. Then, an energy measurement will give sometimes one value, sometimes other values. Once you've measured the energy, you've forced the particle to decide what energy it had: the one you measured. From now on, it is a proper function of the Hamiltonian - at least if your measure is certain enough.

 

Some other operators are compatible (commutative) with the Hamiltonian, and then you can measure the corresponding values together with the energy.

 

Other operators and particle attributes are incompatible with the Hamiltonian (not commutative). Then you can't get fixed measures for both. Measuring the energy, hence freezing it, makes the other attribute uncertain; measuring hence freezing the other attribute makes the energy uncertain.

 

The typical attribute incompatible with energy is time. The proper function of the Hamiltonian, whose energy is certain, are the "stationary" waves, which don't evolve over time. The wave has a term exp(i2pi*Et/h) that lets the phase rotate, but the rest, which gives the amplitude against the position, is independent of time.

 

Because the amplitude is independent of time when the energy is certain, you can't measure "when" for a solution that keeps stationary, the one whose energy is certain. If your experiment tells "when", then after the measure, the state of the particle is a mix of several states of certain energy (or a mix of other states; the ones of certain energy are just one possible choice), the mix that let the particle react at the observed time. After that measure, an energy measure is uncertain, because the state itself is a mix.

 

Uncompatible attributes often relate to an other by a Fourier transform: energy and time, position and momentum... but not always. Take a 2p orbital, here with a peacock shape, the leftmost one

http://winter.group.shef.ac.uk/orbitron/AOs/2p/index.html

its orbital momentum, which is the number of turns of the phase in one geometrical turn around the nucleus, is certain (here zero for the peacock 2p) around the z axis (for the left example). Then the orbital momentum is uncertain around the x and y axes: the phase passes from 0° to 180° in half a geometric turn, but it can do so clockwise or anticlockwise, hence the uncertainty. The peacock around z is a sum of doughnut 2p around x, one clockwise and the other anticlockwise, and the measure sees either one or the other.

 

Particle spin along the axes is incompatible as well, but I can't make my picture of it; its compatibiity works like the orbital momentum, fortunately.

[studiot]

 

Enthalpy

The proper functions of the Hamiltonian are the ones with well-defined energy. If you measure their energy, you get always this one value.

 

The wave can differ from the Hamiltonian's proper ones. Then, an energy measurement will give sometimes one value, sometimes other values. Once you've measured the energy, you've forced the particle to decide what energy it had: the one you measured. From now on, it is a proper function of the Hamiltonian - at least if your measure is certain enough.

 

All values of energy are well defined.

 

The point is that some are accessible and some inaccessible to the system.

These correspond to your 'proper functions'.

[Enthalpy]

You mix the possible waves around a nucleus with the stationary ones.

 

Only the stationary waves, the orbitals, have a well-defined energy, but an electron can have a different shape, which then evolves over time. If the electron is trapped in the atom, it is a weighed sum of orbitals. That weighed sum is a solution to Schrödinger's equation, because the equation is linear.

 

The real difference is "stationary". A weighed sum of orbitals having different energies is not an orbital, because their exp(i2pi*E/h) lets the phases rotate at different speeds, so the sum will strengthen or weaken at places varying with time. The electron moves then, and radiates.

 

In most atoms and molecules, the electrons are stable and fill the lowest obitals. But for instance during the absorption or emission of a photon, electrons are a mix of several orbitals, and their energy is uncertain.

 

The 'proper' functions, or keep the badly translated 'eigen' if you wish, are the orbitals. These are not the only possible shapes of an electron around a nucleus; they are the stable ones.