Eigenvalues and eigenvectors of observables

In summary, the conversation is about calculating the eigenvalues and eigenvectors of a given Hamiltonian using the states |0⟩ and |1⟩ as the basis. The solution involves finding the matrix elements and using them to express the eigenstates in terms of these basis states.
  • #1
Fixxxer125
41
0

Homework Statement



Calculate the Eigenvalues and eigenvectors of
H= 1/2 h Ω ( ]0><1[ + ]1><0[ )

Homework Equations



I know H]λ> = λ]λ>


The Attempt at a Solution


I don't know if I am meant to concert my bra's and ket's into matrices, and if so how to represent these as matrices?
 
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  • #2
Let me make your post look a bit prettier:
Fixxxer125 said:

Homework Statement



Calculate the Eigenvalues and eigenvectors of
$$\hat{H} = \frac{1}{2}\hbar\Omega (|0\rangle\langle 1| + |1\rangle\langle 0| )$$

Homework Equations



I know ##\hat{H}|λ\rangle = λ|λ\rangle##.


The Attempt at a Solution


I don't know if I am meant to concert my bra's and ket's into matrices, and if so how to represent these as matrices?
Are the states ##| 0 \rangle ## and ##| 1 \rangle## orthonormal? If so, just calculate the matrix elements ##\langle i |\hat{H}|j \rangle##.
 
  • #3
Thanks! In the solution given the Eigenstates are given in terms of the |0⟩ and |1⟩ states in the Hamiltonian. How do I know what these states are in terms of matrices so I can write the eigenstates in terms of these? Cheers
 
  • #4
Those two states are the basis you're using, so...
 
  • #5


I would approach this problem by first recognizing that the given operator H is a combination of two matrices, namely ]0><1[ and ]1><0[. These two matrices represent the outer product of the two basis vectors, |0> and |1>, which can be represented as column vectors [1 0] and [0 1], respectively. Therefore, we can write H as:

H = 1/2 h Ω ([1 0][0 1] + [0 1][1 0])

= 1/2 h Ω ([0 1] + [1 0])

= 1/2 h Ω [1 1]

Next, we can use the eigenvalue equation H|λ> = λ|λ> to find the eigenvalues and eigenvectors of H. Plugging in the expression for H, we get:

1/2 h Ω [1 1]|λ> = λ|λ>

Solving for λ, we get two eigenvalues: λ = 1/2 h Ω and λ = -1/2 h Ω.

To find the corresponding eigenvectors, we can substitute these eigenvalues back into the equation H|λ> = λ|λ> and solve for |λ>. For λ = 1/2 h Ω, we get the eigenvector |λ> = [1 1]ᵀ. For λ = -1/2 h Ω, we get the eigenvector |λ> = [-1 1]ᵀ.

In conclusion, the eigenvalues of H are 1/2 h Ω and -1/2 h Ω, and the corresponding eigenvectors are [1 1]ᵀ and [-1 1]ᵀ. These eigenvalues and eigenvectors represent the allowed values and associated states of the observable H.
 

1. What are eigenvalues and eigenvectors of observables?

Eigenvalues and eigenvectors are mathematical concepts used to describe the behavior of a system or observable. In simple terms, eigenvalues represent the possible values that an observable can take on, while eigenvectors represent the corresponding states of the system that produce these values.

2. How are eigenvalues and eigenvectors related to observables in quantum mechanics?

In quantum mechanics, observables are represented by operators that act on the wave function of a system. The eigenvalues of these operators represent the possible outcomes of a measurement of the observable, while the corresponding eigenvectors represent the states of the system that will produce these outcomes.

3. What is the significance of eigenvalues and eigenvectors in quantum mechanics?

Eigenvalues and eigenvectors play a crucial role in quantum mechanics as they provide a way to describe and understand the behavior of physical systems. They allow us to predict the outcomes of measurements and analyze the evolution of a system over time.

4. Can observables have multiple eigenvalues and eigenvectors?

Yes, observables can have multiple eigenvalues and eigenvectors. This means that a single observable can have multiple possible outcomes and corresponding states of the system. For example, the position observable in quantum mechanics can have an infinite number of eigenvalues and eigenvectors.

5. How are eigenvalues and eigenvectors calculated for a given observable?

The process of finding the eigenvalues and eigenvectors of an observable involves solving a mathematical equation known as the eigenvalue equation. The eigenvalues are the solutions to this equation, while the eigenvectors can be found by plugging in the eigenvalues into the equation and solving for the corresponding vector.

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