Question

a) the raising and lowering operators are Hermitian conjugates of each other, that is $\hat{a}_{\pm} = (\hat{a}_{\pm})^{\dagger}$, where $\hat{a}_{\pm} = \frac{1}{\sqrt{2hm\omega}}(\pm i\hat{p} + m\omega\hat{x})$. Prove this. b) Show that for arbitrary functions $f(x)$ and $g(x)$ that $\langle f|\hat{Q}g\rangle = \langle g|\hat{Q}^{\dagger}f\rangle^{*}$. c) The matrix elements of $\hat{Q}$ in a basis are defined as $Q_{mn} = \langle m|\hat{Q}|n\rangle = \langle m|\hat{Q}|n\rangle$ where $|m\rangle$ and $|n\rangle$ are arbitrary basis states. Show how the matrix elements of $\hat{Q}^{\dagger}$ are related to $Q_{mn}$. d) Show that the matrix elements of $\hat{a}_{+}$ and $\hat{a}_{-}$ in the basis of the simple harmonic oscillator energy eigenstates follow the relation you found in c).

          a) the raising and lowering operators are Hermitian conjugates of each other, that is $\hat{a}_{\pm} = (\hat{a}_{\pm})^{\dagger}$, where $\hat{a}_{\pm} = \frac{1}{\sqrt{2hm\omega}}(\pm i\hat{p} + m\omega\hat{x})$. Prove this.
b) Show that for arbitrary functions $f(x)$ and $g(x)$ that
$\langle f|\hat{Q}g\rangle = \langle g|\hat{Q}^{\dagger}f\rangle^{*}$.
c) The matrix elements of $\hat{Q}$ in a basis are defined as $Q_{mn} = \langle m|\hat{Q}|n\rangle = \langle m|\hat{Q}|n\rangle$ where $|m\rangle$ and $|n\rangle$ are arbitrary basis states. Show how the matrix elements of $\hat{Q}^{\dagger}$ are related to $Q_{mn}$.
d) Show that the matrix elements of $\hat{a}_{+}$ and $\hat{a}_{-}$ in the basis of the simple harmonic oscillator energy eigenstates follow the relation you found in c).

a) the raising and lowering operators are Hermitian conjugates of each other, that is â± = (â±)^†, where â± = (1)/(√(2hmω))(± ip̂ + mωx̂). Prove this.
b) Show that for arbitrary functions f(x) and g(x) that
⟨ f|Q̂g⟩ = ⟨ g|Q̂^†f⟩^*.
c) The matrix elements of Q̂ in a basis are defined as Qmn = ⟨ m|Q̂|n⟩ = ⟨ m|Q̂|n⟩ where |m⟩ and |n⟩ are arbitrary basis states. Show how the matrix elements of Q̂^† are related to Qmn.
d) Show that the matrix elements of â+ and â- in the basis of the simple harmonic oscillator energy eigenstates follow the relation you found in c).

Added by Angela A.

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