Question
Show that the operators for the $x$ coordinate and for the momentum in the $x$ direction $p_{x}$ do not commute. Calculate the operator representing the commutator of $x$ and $p_{x}$.
Step 1
In quantum mechanics, the commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as follows: \[ [\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} \] Show more…
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(e) Show that the momentum operators p̂ₓ, p̂ᵧ and p̂_z all commute with each other. (f) The angular momentum operator is defined as L⃗ = r⃗ × p̂ that have components L̂ₓ = ŷp̂_z − ẑp̂ᵧ, L̂ᵧ = ẑp̂ₓ − x̂p̂_z, L̂_z = x̂p̂ᵧ − ŷp̂ₓ Calculate the commutator [L̂ₓ, L̂ᵧ], [L̂ᵧ, L̂_z] and [L̂_z, L̂ₓ]. And what pattern do you find? (g) Define L̂² = L̂ₓ² + L̂ᵧ² + L̂_z². Compute [L̂_z, L̂²]. (h) In class, we have defined Hermitian operators. Is the angular momentum operator Hermitian?
Show that the commutator operator $\hat{A}_{x} \hat{A}_{y}-\hat{A}_{y} \hat{A}_{x}$ is equal to $i \hbar \hat{A}_{z}$.
Find the commutator of the operators (a) $x d / d x$ and $x^{2} d / d x$, (b) energy $\hat{E}$ and time $l$.
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