Calculating <\chi_3|H|\chi_3> w/ Dirac Notation Algebra

However, if you mean |a+ib\rangle = |a\rangle+|ib\rangle, then your original answer was correct.In summary, when given the state |\chi_3\rangle=|a+ib\rangle and the operator H, the resulting expression for <\chi_3|H|\chi_3> depends on the interpretation of |a+ib\rangle. If it is meant to be |a\rangle+i|b\rangle, then the correct expression is <\chi_3|H|\chi_3>= <a|H|a>+<a|H|ib>+<-ib|H|a>+<-ib|H|ib>. However, if it is
  • #1
roro-rose
1
0
Say we have,
|\chi_3>=|a+ib> and we want : <\chi_3|H|\chi_3>

is it correct to say:
<\chi_3|H|\chi_3>= <a|H|a>+<a|H|ib>+<-ib|H|a>+<-ib|H|ib> ??
 
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  • #2
(Hint: learn how do latex on this forum. I've adjusted your stuff below accordingly.
Click on any of the expressions to get more info.)

roro-rose said:
Say we have,
[tex]
|\chi_3\rangle=|a+ib\rangle ~~\mbox{and we want:}~~ \langle\chi_3|H|\chi_3\rangle
[/tex]

is it correct to say:
[tex]
\langle\chi_3|H|\chi_3\rangle ~=~ \langle a|H|a\rangle ~+~ \langle a|H|ib\rangle
~+~ \langle -ib|H|a\rangle ~+~ \langle -ib|H|ib\rangle ??
[/tex]

That's almost right, except that bras are antilinear, hence
[tex]
\langle ib| ~=~ -i \langle b|
[/tex]

So, e.g, your 3rd term should be
[tex]
\langle ib|H|a\rangle ~=~ -i \langle b|H|a\rangle
[/tex]

HTH.
 
  • #3
It really depends on what you mean by [itex]|a+ib\rangle[/itex], which is not a standard notation. Do you mean [itex]|a\rangle+i|b\rangle[/itex]? if so, then strangerep's correction of your result is the right answer.
 

1. What is <\chi_3|H|\chi_3> in Dirac notation algebra?

In Dirac notation algebra, <\chi_3|H|\chi_3> represents the expectation value of the Hamiltonian operator (H) on the state vector |χ₃⟩. This is a way of representing the average value of the energy of the system described by |χ₃⟩.

2. How is <\chi_3|H|\chi_3> calculated using Dirac notation algebra?

To calculate <\chi_3|H|\chi_3> using Dirac notation algebra, we first need to express the Hamiltonian operator (H) in terms of its eigenvalues (Eᵢ) and eigenvectors (|ψᵢ⟩). Then, we can use the formula <\chi_3|H|\chi_3> = ΣᵢEᵢ|⟨χ₃|ψᵢ⟩|², where Σᵢ represents a sum over all possible values of i.

3. What is the significance of calculating <\chi_3|H|\chi_3> in quantum mechanics?

In quantum mechanics, <\chi_3|H|\chi_3> represents the expectation value of the energy of the system described by the state vector |χ₃⟩. This is important because it allows us to make predictions about the behavior and properties of the system based on its energy.

4. Can <\chi_3|H|\chi_3> be negative?

Yes, <\chi_3|H|\chi_3> can be negative. This can happen if the system described by |χ₃⟩ has a higher probability of being in a lower energy state, resulting in a negative expectation value for the energy.

5. Are there any limitations to using Dirac notation algebra to calculate <\chi_3|H|\chi_3>?

One limitation of using Dirac notation algebra to calculate <\chi_3|H|\chi_3> is that it assumes the system is in a pure state, meaning there is no uncertainty in its energy. This may not always be the case in real-world systems, and alternative methods may need to be used to calculate the expectation value.

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