Eigenfunctions and dirac notation for a quantum mechanical system.

In summary, the conversation discusses a quantum mechanical system with a complete orthonormal set of energy eigenfunctions and a corresponding operator \widehat{A}. The system is measured and found to have a value of +1, and the probability of measuring +1 again after a time t is calculated. The conversation also touches on the concept of eigenvectors and eigenvalues in relation to the operator \widehat{A}.
  • #1
Paintjunkie
50
0
QUESTION
A quantum mechanical system has a complete orthonormal set of energy eigenfunctions,
|n> with associate eigenvalues, En. The operator [itex]\widehat{A}[/itex] corresponds to an observable such that
Aˆ|1> = |2>
Aˆ|2> = |1>
Aˆ|n> = |0>, n ≥ 3
where |0> is the null ket. Find a complete orthonormal set of eigenfunctions for
[itex]\widehat{A}[/itex]. The observable is measured and found to have the value +1. The system is unperturbed and then after a time t is remeasured . Calculate the probability
that +1 is measured again.


I would really appreciate any guidance on this. I cannot find this in my book at all. I know my teacher has spoken about it bra-ket notation and such. but it never really made sense. where can I find examples of problems like this but not this.
 
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  • #2
Paintjunkie said:
QUESTION
A quantum mechanical system has a complete orthonormal set of energy eigenfunctions,
|n> with associate eigenvalues, En. The operator [itex]\widehat{A}[/itex] corresponds to an observable such that
Aˆ|1> = |2>
Aˆ|2> = |1>
Aˆ|n> = |0>, n ≥ 3
where |0> is the null ket. Find a complete orthonormal set of eigenfunctions for
[itex]\widehat{A}[/itex]. The observable is measured and found to have the value +1. The system is unperturbed and then after a time t is remeasured . Calculate the probability
that +1 is measured again.


I would really appreciate any guidance on this. I cannot find this in my book at all. I know my teacher has spoken about it bra-ket notation and such. but it never really made sense. where can I find examples of problems like this but not this.

You could start out by telling me what ##{\widehat A}(|1>+|2>)## and ##{\widehat A}(|1>-|2>)## are.
 
  • #3
Aˆ(|1>+|2>) = |2> + |1> ?
Aˆ(|1>−|2>) = |2> - |1> ? I really don't know
 
  • #4
I found this I feel like it should help me but I don't know...

For every observable A, there is an operator [itex]\hat{A}[/itex] which acts upon the
wavefunction so that, if a system is in a state described by |ψ>, the
expectation value of A is
<A>= <ψ|[itex]\hat{A}[/itex]|ψ>= ∫ dx ψ*(x) [itex]\hat{A}[/itex] ψ(x)

that integral is from -∞ to ∞
 
  • #5
ok so I am reading a little more and.

Aˆ(|1>+|2>) = Aˆ|1>+Aˆ|2> ==>α=1, β=1
Aˆ(|1>−|2>) = Aˆ|1>-Aˆ|2> ==>α=1,β=-1

<1|2> = <-2|1>


idk this does not really make sense to me
 
  • #6
Paintjunkie said:
Aˆ(|1>+|2>) = |2> + |1> ?
Aˆ(|1>−|2>) = |2> - |1> ? I really don't know

Yes, A^(|1>+|2>)=|1>+|2> and A^(|1>-|2>)=|2>-|1>=(-1)*(|1>-|2>). What does that tell about eigenvalues and eigenvectors of A^?
 
  • #7
that they are linear and Hermitian

or I guess that's for A^...
 
  • #8
Paintjunkie said:
that they are linear and Hermitian

or I guess that's for A^...

Be more specific. Looking at those two equations I see two eigenvectors of A^ and their corresponding eigenvalues.
 
  • #9
maybe that En is 1 and 2 ?
 
  • #10
Paintjunkie said:
maybe that En is 1 and 2 ?

Noo. x is an eigenvector of A^ if A^(x)=cx for some constant c. What does A^(|1>+|2>)=|1>+|2> tell you?
 
  • #11
is the probability that |1> will give 2 at t=0 |En|2 and that equals 1 ?
 
  • #12
Dick said:
Noo. x is an eigenvector of A^ if A^(x)=cx for some constant c. What does A^(|1>+|2>)=|1>+|2> tell you?

I guess that means that α and β are 1?
 
  • #13
Paintjunkie said:
I guess that means that α and β are 1?

That's still not saying anything about eigenvalues or eigenvectors. Look, A^(|1>+|2>)=|1>+|2> tells me that |1>+|2> is an eigenvector of A^. Why do I say that and what's the corresponding eigenvalue?
 
  • #14
I don't know I give up. thanks for trying man.
 
  • #15
Paintjunkie said:
I don't know I give up. thanks for trying man.

Yeah, something isn't clicking here. But in case you decide to take another crack at it, I'm trying to get you to see that |1>+|2> is an eigenvector of A^ with an eigenvalue of +1 and |1>-|2> is an eigenvector of A^ with an eigenvalue of -1. Review the definition of eigenvector, there's nothing very hard about this part. You are just getting confused by other stuff.
 

1. What is an eigenfunction in quantum mechanics?

An eigenfunction is a mathematical function that represents a possible state of a quantum mechanical system. It is a solution to the Schrödinger equation and has a corresponding eigenvalue, which represents the energy of the system in that state. In quantum mechanics, the state of a system can be described by a linear combination of eigenfunctions.

2. How is dirac notation used in quantum mechanics?

Dirac notation, also known as bra-ket notation, is a mathematical notation used to represent quantum states and operators in quantum mechanics. It uses the symbols |⟩ (ket) and ⟨| (bra) to represent quantum states and their duals, respectively. It allows for a compact and elegant representation of complex calculations in quantum mechanics.

3. What is the significance of eigenfunctions in quantum mechanics?

Eigenfunctions are important in quantum mechanics because they represent the possible states of a system and their corresponding energies. They also form a complete set of basis functions, meaning that any quantum state can be expressed as a linear combination of eigenfunctions. This allows for the analysis and prediction of the behavior of quantum systems.

4. How are eigenfunctions and eigenvalues related?

Eigenfunctions and eigenvalues are closely related in quantum mechanics. An eigenfunction represents a possible state of a system, while its corresponding eigenvalue represents the energy of that state. The eigenvalue is obtained by applying the corresponding operator to the eigenfunction. In other words, the eigenvalue is the result of the measurement of the energy of the system in that state.

5. Can eigenfunctions have complex values in quantum mechanics?

Yes, eigenfunctions can have complex values in quantum mechanics. This is because quantum states can exist in a superposition of multiple states, each with a different probability amplitude. These probability amplitudes are complex numbers, and therefore, the eigenfunctions that represent these states can also have complex values. This allows for a more comprehensive description of the behavior of quantum systems.

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