Quantum Chemistry 3.3 - Eigenvalues and Eigenfunctions

Short lecture on eigenvalues and eigenfunctions.

When the result of an operator acting on a function is a constant multiplied by the same function, the function is called an eigenfunction, and the constant is called an eigenvalue. The eigenvalue problem is to determine the set of eigenfunctions and eigenvalues for a given operator. The Schrodinger equation is an eigenvalue problem for the Hamiltonian operator, where the wavefunction is the eigenfunction and the eigenvalue is the total energy of the state. Eigenvalue equations could be solved in quantum mechanics for any operator, including kinetic energy, potential energy, momentum, and position.

--- About TMP Chem ---

All TMP Chem content is free for everyone, everywhere, and created independently by Trent Parker.

--- Video Links ---

--- Social Links ---

--- Equipment ---

Microphone: Blue Yeti USB Microphone

Drawing Tablet: Wacom Intuos Pen and Touch Small

Drawing Program: Autodesk Sketchbook Express

Screen Capture: Corel Visual Studio Pro X8

[maddio]

I have never heard a more clear, concise explanation of eigenvalues and eigenfunctions! Thank you for the great video!!

[TheExperiment]

For many months, I've searched textbooks, YouTube, lecture notes and online resources for the perfect quantum mechanics lesson for me to self study in my army days. So far, you're the best at what you do, so keep it up :) your videos are very informative and extremely helpful, not glossing over the foundations many resources take for granted. I had a great time learning from you and your videos. Thank you so much!!!

[PunmasterSTP]

Eigenvalues and eigenfunctions? More like "Amazing videos that you should never shunned!"

[nupurgupta]

Sir I have understood the eigenfunction and eigenvalue thanks from India 🇮🇳

[lapertica]

better than any books. congratulations

[mortezakhoshbin]

im so excited for finding your channel. you explain amazing. so simple explanation you give and others makes it more difficult. thank you. god bless you

[sunitgautam]

So nice to have all the commonly used operators summarized. Nice video! :)

[scottford]

Just started learning this in lecture. Thank you.

[jiyoonha]

I have several questions. Why do classical mechanics properties need quantum operators? What can we learn from these quantum operators? and why does the quantum momentum operator in the x-direction have a minus i value (why not plus i?)

[connyceca]

Hey, your videos are great. This course saves my grades^^ I am studying Biochemistry in Munich and i have quantum mechanics in this semester and i don't understand anything. It's horrible. but i think with the help of your great videos i will pass the test. Basically awesome and a great THANK you from Germany :)

[adityakolhatkar]

Hi! I love your videos! They're simply great! I'm in high school and I'm trying to study quantum mechanics on my own since I'm very interested. Luckily, I am also well acquainted with calculus. Your videos help a *lot*.
Could you please tell me why we take the negative square root for the momentum operator?

[kalyanimehta]

is H^ eigen function and E is eigen value here?

[kalyanimehta]

why do we require eigenvalues and eigen function
?

[ashmatthew]

Sir, shouldn't there be a positive coefficient on the (h bar)^2 d^2/dx^2 term, because -i squared is positive 1.

[nkyu]

in comment, you mentioned u(x, t) = A * exp(i[kx - wt]) as the wave function. this function confuses me a bit. it's somewhat familiar but I'm not sure exactly where it came from.

 -First approach : assuming this is a kind of standing wave we've been dealing with, I set up this function u(x, t)=B(sin(kx-wt)+sin(kx+wt)) = ? = A * exp(i[kx - wt])
-Second approach : u(x, t) = X(x)T(t) = Ccos(~)sin(~)= ? = A * exp(i[kx - wt])

maybe i'm being too serious for things you've just randomly chosen -_-;;

[narayansingh]

Hi, I had a quick question: is there any specific reason why the potential operator is not a part of the momentum operator (given that p^2 = 2m ( E - V(x)) ?

[yanqingliu]

Hi! Thank you for your fantastic lectures! For the momentum operator, I found it confused because if p^2==-hbar^2*D^2, should be p==i*hbar*D. Could you please have a look? I realized that when I was watching average momentum computation.

[nkyu]

using concept of commutator, it seems T_hat and p_hat have same eigenfunction.
likewise for x_hat and V_hat? and for Hamiltonian, is it alone?

if these operators doesn't have same eigenfunction like this,
why do T_hat and V_hat act like they have same eigenfunction?
i mean in time-independent Schrodinger Eqn, T_hat Ψ+V_hat Ψ = EΨ = TΨ + VΨ,
it looks like these two operators have same eigenfunction though they do not commute.

[MonuG]

Can the eigen value of a wave function be negative in quantum mechanics or not

[manuelsojan]

i thought operators were multiplied with a function f(x)...for the kinetic and potential energy operators, what function are they multiplying? f(x) = 1?