Finding the Ground State of a Hamiltonian Operator

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To find the ground state of a Hamiltonian operator represented by a 3x3 matrix, one must identify the eigenvalue corresponding to the lowest energy, which is 0 in this case. The ground state can indeed have zero energy, but the physical implications depend on the potential involved. The kinetic energy cannot be zero, as there should always be some motion even in the ground state. To calculate the uncertainty relation, one must compute the commutator [H,A] and find the eigenvector associated with the ground state eigenvalue. If the expectation value of the commutator in the ground state is zero, it may indicate an issue with the calculations or assumptions made.
Matthollyw00d
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When given a Hamiltonian operator (in this case a 3x3 matrix), how do you go about find the ground state, when this operator is all that is given? By the SE when have H\Psi=E\Psi. I can easily solve for Eigenvalues/vectors, but which correspond to the ground state, or am I missing something?
 
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The ground state is the state with the lowest energy.
 
So I achieve eigenvalues of \{0,1,2\} (this is actually the eigenvalues I have for the problem), would the 0 or the 1 be the lowest state? I assumed 0, but can a ground state have 0 energy?
 
It depends on the potential. You can always add a constant to the potential without changing anything physically, which would shift the total energy by the same amount. What you can't really have is the kinetic energy being 0 because there should always be some motion even in the ground state.
 
Let H=\hbar\omega \[ \left( \begin{array}{ccc}<br /> 1 &amp; i &amp; 0 \\<br /> -i &amp; 1 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1 \end{array} \right) \]
and let A =\hbar \left[ \begin{array}{ccc}<br /> 1 &amp; 0 &amp; i \\<br /> 0 &amp; 1 &amp; 0 \\<br /> -i &amp; 0 &amp; 1 \end{array} \right].

Calculate the uncertainty relation \sigma_E \sigma_a for a system in the energy ground state.

My problem is calculating \langle [H,A]\rangle.
The commutator [H,A]=\hbar^2\omega \left[ \begin{array}{ccc}<br /> 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1 \\<br /> 0 &amp; -1 &amp; 0 \end{array} \right]
And if relevant the eigenvalues of H are \hbar\omega \{0,1,2\}.
It says to calculate in the ground state, but I don't know the ground state, so how do I gather it from this data?
 
You have the eigenvalue for the ground state, now find the corresponding eigenvector. It will be a (non-zero) vector

v_0 = \begin{pmatrix} a \\ b \\ c \end{pmatrix}

satisfying

H v_0 = 0.
 
Yes, but \langle v_0|[H,A]|v_0 \rangle =0 which gives me a trivial inequality, which leads me to believe something is incorrect and hence why I posted here.
 
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