Eigenvalues and ground state eigenfunction of a weird Hamiltonian

[Thunder_Jet]

Hello again everyone!

I would like to ask a question regarding this Hamiltonian that I encountered. The form is H = Aa^+a + B(a^+ + a). Then there is this question asking for the eigenvalues and ground state wavefunction in the coordinate basis. The only given conditions are, the commutator of a^+ and a is [a^+,a] = 1, and that A > 0 and B are c-number constants. I actually do not understand the meaning of c-number constants. Can anyone suggest how to attack this problem?

 

[vanhees71]

Are you sure? The canonical convention is that $\hat{a}$ is the destruction and $\hat{a}^{\dagger}$ the creation operator for phonons. Then the commutator should read $[\hat{a},\hat{a}^{\dagger}]=1$. What you have here, is simply a "shifted harmonic oscillator". This problem you find solved in many textbooks on quantum mechanics.

 

[Thunder_Jet]

Are you sure? The canonical convention is that $\hat{a}$ is the destruction and $\hat{a}^{\dagger}$ the creation operator for phonons. Then the commutator should read $[\hat{a},\hat{a}^{\dagger}]=1$. What you have here, is simply a "shifted harmonic oscillator". This problem you find solved in many textbooks on quantum mechanics.

Ok, so a and a^+ are the annihilation and creation operators in the harmonic oscillator problem. I thought there are other operators. Thanks for your comment! Anyway, in this shifted harmonic oscillator case, do you expect that the solution for example, the eigenvalues are just shifted by a constant? I think the same is true for the wavefunction.

 

[Thunder_Jet]

May I know how can I obtain the eigenvalues using the usual eigenvalue problem here? I am quite confused here now.

 

[Avodyne]

Define new creation and annihilation operators
$$\tilde a = a+c$$
$$\tilde a^\dagger = a^\dagger + c$$
where $c$ is a real constant. Choose $c$ so that the hamiltonian is
$$H=A \tilde a^\dagger \tilde a + d$$
where $d$ is another constant. Note that
$$[\tilde a, \tilde a^\dagger]=1$$

 

[Thunder_Jet]

Define new creation and annihilation operators
$$\tilde a = a+c$$
$$\tilde a^\dagger = a^\dagger + c$$
where $c$ is a real constant. Choose $c$ so that the hamiltonian is
$$H=A \tilde a^\dagger \tilde a + d$$
where $d$ is another constant. Note that
$$[\tilde a, \tilde a^\dagger]=1$$

Hmmm, sounds ok. Thank you for your suggestion. But I am really new to ladder operators, how would you use this translated a+ and a?