{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Completeness and time - eigenfunctions(10.9(The functions...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Completeness and time-dependence In the discussion on formal aspects of quantum mechanics I have shown that the eigenfunctions to the Hamiltonian are complete, i.e., for any (10.5) where (10.6) We know, from the superposition principle, that (10.7) so that the time dependence is completely fixed by knowing at time only! In other words if we know how the wave function at time can be written as a sum over eigenfunctions of the Hamiltonian, we can then determibe the wave function for all times. Simple example The best way to clarify this abstract discussion is to consider the quantum mechanics of the Harmonic oscillator of mass and frequency , (10.8) If we assume that the wave function at time is a linear superposition of the first two
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: eigenfunctions, (10.9) (The functions and are the normalised first and second states of the harmonic oscillator, with energies and .) Thus we now kow the wave function for all time: (10.10) In figure 10.1 we plot this quantity for a few times. Figure 10.1: The wave function ( 10.10 ) for a few values of the time . The solid line is the real part, and the dashed line the imaginary part. The best way to visualize what is happening is to look at the probability density, (10.11) This clearly oscillates with frequency . how that . Another way to look at that is to calculate the expectation value of : (10.12) This once again exhibits oscillatory behaviour!...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern