• Support PF! Buy your school textbooks, materials and every day products Here!

Prove the Hamiltonian Operator is Hermitian

  • Thread starter atay5510
  • Start date
  • #1
10
0

Homework Statement



Show that the Hamiltonian operator ([itex]\hat{H}[/itex])=-(([itex]\hbar[/itex]/2m) d2/dx2 + V(x)) is hermitian. Assume V(x) is real

Homework Equations



A Hermitian operator [itex]\hat{O}[/itex], satisfies the equation

<[itex]\hat{O}[/itex]>=<[itex]\hat{O}[/itex]>*

or

∫[itex]\Psi[/itex]*(x,t)[itex]\hat{O}[/itex][itex]\Psi[/itex](x,t)dx = ∫[itex]\Psi[/itex](x,t)[itex]\hat{O}[/itex]*[itex]\Psi[/itex]*(x,t)dx between -∞ and +∞


The Attempt at a Solution



This is my first time using LaTex and I'm having trouble inputting what I want, but basically, I've just substituted the Hamiltonian operator for [itex]\hat{O}[/itex], into the second expression above. This is where I am stuck. I'm essentially left with two parts, one thats "telling" me to prove that the Kinetic Energy portion of H is Hermitian and another telling me to prove the Potential Energy portion is hermitian. However I can't seem to see how to do it.

Thanks

P.s. can you help with a solutio that does not use Bra-Ket notation, this question is in the section that precedes Dirac notation
 

Answers and Replies

  • #2
Simon Bridge
Science Advisor
Homework Helper
17,847
1,644
  • #3
dextercioby
Science Advisor
Homework Helper
Insights Author
12,981
540
Working in the Hilbert space L2(R) one proceeds like this:

a) finds the domain of H.
b) checks if domain is dense everywhere in H.
c) finds the domain of [itex] H^{\dagger} [/itex]
d) checks that the domain of H is included in the domain of its adjoint.
e) finally checks that the ranges of the 2 operators are equal for all vectors in the common domain (the domain of H).
 

Related Threads on Prove the Hamiltonian Operator is Hermitian

Replies
1
Views
2K
Replies
4
Views
346
Replies
1
Views
5K
  • Last Post
Replies
1
Views
2K
Replies
22
Views
31K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
17
Views
4K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
10
Views
2K
Top