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Question 2: Generalized Uncertainty Principle (G.U.P.) a. Consider a 2-dimensional Hilbert space spanned by orthonormal basis vectors $|1\rangle \approx \begin{pmatrix} 1 \ 0 \end{pmatrix}$, $|2\rangle \approx \begin{pmatrix} 0 \ 1 \end{pmatrix}$. In this basis, two operators are represented by $\hat{S_x} \approx \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$ and $\hat{S_y} \approx \frac{\hbar}{2} \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}$. Show that the G.U.P. is satisfied for these two operators when the system is in state $|1\rangle$. b. Show that the G.U.P. is satisfied when $\hat{A} = \hat{x}^2$, $\hat{B} = \hat{p}$, and the state of the system is represented by the position space wavefunction $\psi(x) = Ne^{-ax^2/2}$, where $N$ is a normalization constant.

          Question 2: Generalized Uncertainty Principle (G.U.P.)
a. Consider a 2-dimensional Hilbert space spanned by orthonormal basis vectors
$|1\rangle \approx \begin{pmatrix} 1 \ 0 \end{pmatrix}$, $|2\rangle \approx \begin{pmatrix} 0 \ 1 \end{pmatrix}$. In this basis, two operators are represented by $\hat{S_x} \approx \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$ and $\hat{S_y} \approx \frac{\hbar}{2} \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}$. Show that the G.U.P. is satisfied for these two
operators when the system is in state $|1\rangle$.
b. Show that the G.U.P. is satisfied when $\hat{A} = \hat{x}^2$, $\hat{B} = \hat{p}$, and the state of the system is
represented by the position space wavefunction $\psi(x) = Ne^{-ax^2/2}$, where $N$ is a
normalization constant.

Question 2: Generalized Uncertainty Principle (G.U.P.)
a. Consider a 2-dimensional Hilbert space spanned by orthonormal basis vectors
|1⟩≈
< p m a t r i x >, |2⟩≈
< p m a t r i x >. In this basis, two operators are represented by Ŝx̂≈(ħ)/(2)
< p m a t r i x > and Ŝŷ≈(ħ)/(2)
< p m a t r i x >. Show that the G.U.P. is satisfied for these two
operators when the system is in state |1⟩.
b. Show that the G.U.P. is satisfied when Â = x̂^2, B̂ = p̂, and the state of the system is
represented by the position space wavefunction ψ(x) = Ne^-ax^2/2, where N is a
normalization constant.

Added by Charles C.

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