1
$\begingroup$

Hi I am new to solid state physics and am reviewing a prior knowledge section and would like some clarification. The following appeared in the course notes:

enter image description here

From my understanding, Eigenstates are only eigenstates when associated with the appropriate operator. Are the terms Eigenfunction and Eigenstate analogous? I understand the concept of Eigenfunctions such as e^(3x) but am having a hard time matching the terminology to the physics and so would be grateful for some help <3

Improve this question
$\endgroup$
6
  • $\begingroup$ Quantum harmonic oscillator. $\endgroup$
    – march
    Commented Apr 11, 2023 at 15:21
  • $\begingroup$ cheers I will check this out! $\endgroup$
    – Harry J Critchfield
    Commented Apr 11, 2023 at 15:24
  • $\begingroup$ Have you taken a quantum mechanics course? $\endgroup$
    – Roger V.
    Commented Apr 11, 2023 at 15:31
  • $\begingroup$ Yes I have but am undergrad and revising it for the first time for upcoming exams. Any useful resources would be greatly appreciated! $\endgroup$
    – Harry J Critchfield
    Commented Apr 11, 2023 at 15:49
  • 1
    $\begingroup$ @HarryJCritchfield eigenfunctions, eigenstates, and eigenkets are essentially equivalent, but Hilbert space is not: Hilbert space is the vector space consisting of all wavefunctions/states/kets, not just the set of eigen-functions/states/kets. $\endgroup$
    – Jahan Claes
    Commented Apr 11, 2023 at 16:33

1 Answer 1

Reset to default
1
$\begingroup$

The term eigenfunction and eigenkets are as different as kets and wave function, or say matrices and operators, i.e to say they essentially have the same content, but just appear different in presentation(A fancy way to say this is to say that they are isomorphic). Recall that you could represent a Hermitian Operator by a matrix.Same goes in this case, eigekets or kets in general have representations as wave functions.

Improve this answer
$\endgroup$
5
  • $\begingroup$ Thank you for this answer! Ok so my next question is in what context are the different presentations more useful? When would someone decide to use an eigenket rather than an eigenfunction? Are there some examples I could read up on? Cheers $\endgroup$
    – Harry J Critchfield
    Commented Apr 11, 2023 at 18:58
  • $\begingroup$ When you want to solve differential equations(Schrodinger eqn, for instance) visualise the probability amplitudes, you do the wave function thing. Ket vectors are abstract, however you can associate to each ket a wave function. Recall from classical Mechanics you could do projections of a vector along x, y, z axis. Wavefunction is just like that, the projection of a state vector along the eigenkets. For example consider a system with a definite position, the corresponding wave function is the dirac delta, I. E only the projection along a particular eigenket has value, but the others are 0. $\endgroup$
    – Cbb Ttt
    Commented Apr 11, 2023 at 20:06
  • $\begingroup$ Please not that the explanation that I gave you in the comments was meant to aid your understanding, it was, however, not a completely precise answer, for that you need to refer to a good book, I recommend, the book called "A Modern Approach to Quantum Mechanics" authored by John. S. Townsend. For a bit more advanced treatment, refer to J. J. Sakurai's Modern Quantum Mechanics. $\endgroup$
    – Cbb Ttt
    Commented Apr 11, 2023 at 20:25
  • $\begingroup$ Thanks for all the help greatly appreciated whilst my uni lecturers are ignoring emails! $\endgroup$
    – Harry J Critchfield
    Commented Apr 12, 2023 at 9:39
  • $\begingroup$ Do remember again, the explanation given to you in the comments is only approximately true, I did so because I though it might not be a great idea to give you a sophisticated/abstract answer at this point of your learning, you should refer to good books, you should do that. $\endgroup$
    – Cbb Ttt
    Commented Apr 12, 2023 at 9:53

Not the answer you're looking for? Browse other questions tagged or ask your own question.