No, energy levels are not always discrete. The interactions among the atoms in a single-crystal solid, for example, allow for continuous ranges.

Discrete energy levels of a quantum system is of course an idealization. To "see them" you perturb them "appropriately". The perturbing operator decides according to specific sum rules (depending on the operator) what transitions are possible, their intensities and halfwidths (or reciprocally their life time).

The full hamiltonian, including the perturbing electromagnetic field has of course a continuous spectrum, although the (previously) discrete states are now resonances in a continuous energy spectrum.

Good point. It's kind of hard to talk about the discrete energy of a system that resides in Hilbert space with no known one-to-one correspondence (in terms of measurable properties) with the "real world". The orthodox interpretation implies that the question asks about the physical properties of systems described by an irreducibly statistical mechanics.

That said, it's still important (experimentally, theoretically, and most of all computationally) to distinguish between discrete and continuous spectra, regardless of the interpretation of measurement or of quantum mechanics itself.

In the simplest case, atomic energy levels for free atoms are discrete when the electron is bound and its energy is negative. If however the electron is not bound, its energy is positive and can take continuous values. Interaction may always add other dimensions.

I think the question Manoj arises is related to bound states of negative energy (for example, atomic or molecular orbitals) which present a discrete set of energy levels.

Manoj asks for a possible mutual links of these levels. The answer is "yes", these levels are characteristic of the forces involved in that bond. Therefore, there are not independent from one other, but instead different aspects of the same thing: the structure of the binding force.

Technically, energy levels are the eigenvalues of the Hamiltonian operator associated with those forces.

Maybe a clarification is needed.

The spectrum of a selfadjoint operator is usually decomposed into the pure point part, the absolutely continuous- and the singularly continuous part. Fortunately for us chemical phycists atomic and molecular Coulomb Hamiltonians do not have a singularly continuous part, the latter with important consequences for the establishment of resonance states and their lifetimes.

Pioneering QM does not in itself "permit" quantum jumps, however, the atom (or molecule etc.) being bound, i.e. finding itself in the pure point spectrum, is as said a fiction. In general an excited state will spontaneously return to a lower state by emitting a photon. This implies that such discrete states have a natural broadening and a corresponding life time, commensurate with the uncertainty principle.

Interacting with the photonic environment imparts an additional perturbing operator to the Hamiltonian. As we all know an atom in an electromagnetic field changes all its bound states into resonances. Fortunately these effects can be determined as very small, usually handled by the Fermi Golden rule and a Lorentzian ansatz.

Even if these techniques are well understood and experimentally and theoretically investigated and analysed, it is important to remember that the "small" perturbation added to the atomic system imparts a fundamental change of the spectral properties of the Hamiltonian, but this is another story.

For sake of precision, my remark was referred to the first quantization of an isolate atom or molecule (we say). In general these systems are coupled with an external field which can be, in addition, quantized (second quantization). In this wider scenario "discrete" levels are effectively resonances.

Thankyou for all these answers.But I want to ask that will the electron go instantaneously from one energy level to another level or it will take some time .If it takes finite times then we can not say that energy levels are discrete.

Dear Manoj,

Here are two examples:

1. The D1 transition in the rubidium atom, Rb. Emitting the photon takes 26ns (1ns=10-9s) corresponding to an 8m long photon.

2. The hyperfine transition of cesium 133 (133Cs) corresponds to a microwave frequency of 9.2•109 Hz. Here the lifetime is about 1000s corresponding to a length of 3•1011m. (this is the present definition of the time unit in atomic clocks, which however is going to be dramatically improved).

 To

Prof. Brändas sir,

the time you have mentioned in your answer is the life time of being in the excited state

I am asking the transition time , that is the time taken to go from exited to ground state.If it has some finite transition time then it has to pass  between the allowed energy levels but then it will have the energy in between the two energy levels which is not permitted.

Dear Manoj,

The lifetime of the excited state that I have mentioned depends on the spontaneous emission of the photon and its "qualities "(the length of the photon and its frequency). These facts determine also the lower state the system will find itself in after the transition.

In this interpretation we have not allotted any specific time for describing the actual transition, since the latter has not been defined. This means that if there are several transitions the lifetimes that we determine should be related to the actual times the system spends in each intermediate state.

I believe this definition gives more sense than taking the lifetimes, described above, to, instead of interpreting that a lifetime imparts the time spent in a certain level, that the same time should be defined as the dynamic process of the system making the actual transition. Poineering QM does not tell which one to believe.

The point I want to make is that there are not at the present understanding of physics any other times involved here than what I have quoted, unless you are able to assign additional perturbations to the Hamiltonian making the process more complex.

The evolution of the wave function is a continuios process resolved in time. Modern methods of attoscecond laser probing allow the tracing of this evolution in the course of an electronic transition. However, the state of the system during this evolution is not stationary and, hence, is not characterised by a definite energy. Each energy measurement yields an eigenvalue (level) of energy with a certain time-dependent probability.

V.L. Derbov,

This is exactly my point! One can note the rather intense discussions over time regarding the impossibility to account for quantum jumps in standard QM.

The situation is quite different when considering novel extensions to non-Hermitian QM developed recently in chemical physics.

Dear Dr. Manoj Das,

Your question regarding detection of exact time at which an electron makes a transition from an upper energy  level to a lower level, reminds me of the interest taken by the quantum optics community about three decades ago. It was suggested that if an electron is making  transitions from upper level, say, 1 to a lower level, say, 2 we may connect the level 2 to some other level 3 by an exactly resonant intense laser. Whenever the electron is in state 2, it keeps on oscillating between the levels 2 and 3 because of the Rabi Phenomenon and fluorescent radiation of the laser frequency may be seen. However, when the electron exists in level 1, no fluorescence is seen. Thus one can find exactly when the electron is in level 1 and when it is in level 2 (of course, making Rabi oscillations between levels 2 and 3).

One may attempt an extension of this idea  where the level 1 is also connected with such a level  4 which is not connected to any of the other levels. Presence of electron in level 1 (connected with level 4 by Rabi Oscillations) will then be detected by the fluorescence of the second intense laser.

Reference to the paper discussed is MS Kim & PL Knight: Quantum-jump telegraph noise and macroscopic intensity fluctuations

Phys. Rev. A 36, 5265 – Published 1 December 1987

http://nvlpubs.nist.gov/nistpubs/Legacy/NSRDS/nbsnsrds35v1.pdf

https://www.gpo.gov/fdsys/pkg/GOVPUB-C13-e4ed35e343832d00ddc6159bc4edcb25/pdf/GOVPUB-C13-e4ed35e343832d00ddc6159bc4edcb25.pdf

https://www.colorado.edu/physics/phys2170/phys2170_fa08/Lecture_Notes/class20_AtomicEnergyLevels.pdf