Learning Dirac Notation: Writing Hamiltonian for 3 States

[noman3k3]

I am new to quantum physics. My question is how to write the Hamiltonian in dirac notation for 3 different states say a , b , c having same energy.

I started with eigenvalue problem H|Psi> = E|psi>

H = ? for state a?

SO it means that indvdually H= E (|a><a|) for state a
and for all three states i can write
H= E (|a><a|+|b><b|+|c><c|)

am i right?

 

[dextercioby]

H = E(|a><a|+...) would be a hint.

 

[noman3k3]

SO it means that indvdually H= E (|a><a|) for state a
and for all three states i can write
H= E (|a><a|+|b><b|+|c><c|)

am i right?

 

[A. Neumaier]

I am new to quantum physics. My question is how to write the Hamiltonian in dirac notation for 3 different states say a , b , c having same energy.

I started with eigenvalue problem H|Psi> = E|psi>

H = ? for state a?

SO it means that indvdually H= E (|a><a|) for state a
and for all three states i can write
H= E (|a><a|+|b><b|+|c><c|)

am i right?

The H you want is most likely a 3x3 matrix. If your states a, b c are orthogonal eigenstates, the energies are the diagonal entries and the off-diagonal entries were zero. In Dirac notation, this is
H= E_a|a><a|+E_b|b><b|+E_c|c><c|

But only if the eignestates are orthogonal and eigenstates!

 

[noman3k3]

So if states are degenerate i can have

H= [E_a 0 0 psi = [a
0 Eb 0 b
0 0 EC] c]

 

[A. Neumaier]

So if states are degenerate i can have

H= [E_a 0 0 psi = [a
0 Eb 0 b
0 0 EC] c]

Your equation looks garbled. Please use an intelligible format, to be able to answer your question.
(Probably the answer to your question is yes, but since the question isn't clear, the answer cannot be.)

 

[noman3k3]

H= [E_a 0 0 ; 0 E_b 0; 0 0 E_c]

Psi= [a; b;c]

 

[A. Neumaier]

H= [E_a 0 0 ; 0 E_b 0; 0 0 E_c]

Psi= [a; b;c]

Yes, this is meaningful. The formula for H says that you represent the Hamiltonian in an eigenbasis, and the formula for psi says that the state you consider decomposes in this eigenbasis with coefficients a, b, and c.