QP: Physical Meaning of Orthogonality

[WWCY]

Homework Statement

I have recently come across the notation <ψ|Φ> in my notes and am not quite sure what it means. Some articles I have read online state that this is analogous to the dot product, except that this is the "dot-product" of 2 wave-functions.

Would I then be right in saying that "ψ" can be taken as a bunch of possible eigen-states in a system, and performing ( <ψ|Φ> )² provides me with the probabilities of measuring state Φ with its corresponding eigen-energy in that system?

I really do apologize if my question isn't phrased coherently as this is (very literally) all greek to me.

Assistance and clarification is greatly appreciated.

Homework Equations

The Attempt at a Solution

 

[DrClaude]

I have recently come across the notation <ψ|Φ> in my notes and am not quite sure what it means. Some articles I have read online state that this is analogous to the dot product, except that this is the "dot-product" of 2 wave-functions.

Yes, this is how the scalar product is represented in the Dirac notation.

Would I then be right in saying that "ψ" can be taken as a bunch of possible eigen-states in a system, and performing ( <ψ|Φ> )² provides me with the probabilities of measuring state Φ with its corresponding eigen-energy in that system?

That is correct, but only if $\phi$ is an energy eigenstate. (And that should be $| \langle \psi | \phi \rangle|^2$, with vertical lines representing the absolute value).

 

[WWCY]

Thank you so much for the help. Could you also briefly explain what you mean by an energy eigenstate?

 

[DrClaude]

Could you also briefly explain what you mean by an energy eigenstate?

An eigenstate of the Hamiltonian.

Given
$$
\hat{H} | n \rangle = E_n | n \rangle
$$
then $| \langle n | \psi \rangle|^2$ will given you the probability of measuring the energy of a system in state $| \psi \rangle$ as $E_n$. But $| \langle \phi | \psi \rangle|^2$ is the generic scalar product, with $\phi$ representing any arbitrary state.

 

[WWCY]

An eigenstate of the Hamiltonian.

Given
$$
\hat{H} | n \rangle = E_n | n \rangle
$$
then $| \langle n | \psi \rangle|^2$ will given you the probability of measuring the energy of a system in state $| \psi \rangle$ as $E_n$. But $| \langle \phi | \psi \rangle|^2$ is the generic scalar product, with $\phi$ representing any arbitrary state.

I guess i'll need to do more reading on this then, thank you!

 

[WWCY]

Hi DrClaude, if both Ψ and Φ represented a superposition of eigenstates, what would the physical interpretation of |<Ψ|Φ>|² be?

Also, if i have |Ψ> = i/6 |Φ₁ + 1/√6 |Φ₂ + 2/√6 |Φ₃,

what do the fractions beside the wavefunctions Φ represent?

 

[PeroK]

Hi DrClaude, if both Ψ and Φ represented a superposition of eigenstates, what would the physical interpretation of |<Ψ|Φ>|² be?

Also, if i have |Ψ> = i/6 |Φ₁ + 1/√6 |Φ₂ + 2/√6 |Φ₃,

what do the fractions beside the wavefunctions Φ represent?

How and where are you learning QM?

Your questions suggest you have just jumped in at a random point and are confused by all you see around you!

 

[WWCY]

To be fair, I am very much confused. I'm getting stuff off of my school notes which are patchy at best and so I'm trying to fill in the gaps myself. There's a lot of "put this into this equation and get this" but it's the "what does this and that mean" I'm really struggling with.

Any help and advice is greatly appreciated!

 

[PeroK]

To be fair, I am very much confused. I'm getting stuff off of my school notes which are patchy at best and so I'm trying to fill in the gaps myself. There's a lot of "put this into this equation and get this" but it's the "what does this and that mean" I'm really struggling with.

Any help and advice is greatly appreciated!

You need some coherent material from somewhere. You didn't really answer my question. I guess you are not using a textbook?

 

[WWCY]

Nope. The textbook is used in QM I, but the course I'm doing is sort of an introductory QM course. I tried flipping through the text but the concepts were much too advanced for the material i have in my set of notes.

 

[PeroK]

Nope. The textbook is used in QM I, but the course I'm doing is sort of an introductory QM course. I tried flipping through the text but the concepts were much too advanced for the material i have in my set of notes.

It's not really possible through this forum to teach you QM from the ground up. That's what textbooks are for. Have you talked to whomever is running the course?

 

[WWCY]

I haven't, but I'm intending to as the notes are getting harder to understand.

Is it possible though, for you to provide a watered down explanation as to what the notation I wrote means?

 

[PeroK]

I haven't, but I'm intending to as the notes are getting harder to understand.

Is it possible though, for you to provide a watered down explanation as to what the notation I wrote means?

This one?

Hi DrClaude, if both Ψ and Φ represented a superposition of eigenstates, what would the physical interpretation of |<Ψ|Φ>|² be?

Also, if i have |Ψ> = i/6 |Φ₁ + 1/√6 |Φ₂ + 2/√6 |Φ₃,

what do the fractions beside the wavefunctions Φ represent?

 

[PeroK]

PS Here is something I found on t'Internet:

https://www-thphys.physics.ox.ac.uk/people/JamesBinney/qb.pdf

Looks not bad for a free PDF. Might be worth a read. Section 1.3 covers the Dirac notation.

 

[WWCY]

This one?

Yup, the one you quoted. And thank you for the PDF, i'll give it a good readthrough.

 

[PeroK]

Hi DrClaude, if both Ψ and Φ represented a superposition of eigenstates, what would the physical interpretation of |<Ψ|Φ>|² be?

Also, if i have |Ψ> = i/6 |Φ₁ + 1/√6 |Φ₂ + 2/√6 |Φ₃,

what do the fractions beside the wavefunctions Φ represent?

That's a vector equation, with $|\psi \rangle$ representing the vector (ket in Dirac notation). These vectors (kets) represent the state of the system (the wave function in wave mechanics). You may remember things like:

$\vec{v} = 3 \vec{i} + 2 \vec{j} - 6 \vec{k}$

It's the same idea, with $|\phi_1 \rangle, |\phi_2 \rangle, |\phi_3 \rangle$ as three orthonormal basis kets (in this case three from an infinite set, as the state space is infinite dimensional).

 

[DrClaude]

It seems to me that you need to revise linear algebra.