Recent content by Sofie RK

Homework Statement I need calculate the points (##x_i##) and weights (##w_i##) with Gauss Lobatto seven points on the interval [a,b]. With the points and the weights I am going to approximate any integral at this interval.Homework Equations I have found the relevant points and weights at the...

What about this: $$\sigma^2 = \big \langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)^2 \psi \big\rangle$$ $$\sigma^2 = 0 \Rightarrow \big \langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)^2 \psi \big\rangle = 0 $$ Hermiticity: $$ \big \langle (\hat{H} - \langle\hat{H}\rangle) \psi \mid...

Suggestion:$$ \Big\langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)(\hat{H} - \langle\hat{H}\rangle)\psi \Big\rangle = \Big\langle (\hat{H} - \langle\hat{H}\rangle)\psi \mid (\hat{H} - \langle\hat{H}\rangle)\psi \Big\rangle $$ $$ = \Big\langle \hat{H}\psi - \langle\hat{H}\rangle\psi \mid...

Hermitian operators have real eigenvalues and orthogonal eigenfunctions. $$(\hat{H} - \langle \hat{H} \rangle)\psi = \lambda\psi,$$ where ##\lambda## is a real number. Is this something I can use? Then, $$ \sigma^2 = \big\langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)(\hat{H} -...

##\hat{H} - \langle\hat{H}\rangle ## is a hermitian operator, which means that its eigenvalues must be real. But I don't understand what happens when a hermitian operator is squared.

Homework Statement For a wavefunction ##\psi##, the variance of the Hamiltonian operator ##\hat{H}## is defined as: $$\sigma^2 = \big \langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)^2 \psi \big\rangle$$ I want to prove that if ##\sigma^2 = 0##, then ##\psi## is a solution to the...

Thank you! This really helped!

Okey, then the integrals will be the same. But if ##x_1## refers to particle 1 and ##x_2## refers to particle 2, why can I choose that ##x_1 = X## in the first integral, ##x_1 = Y## in the second integral, and then compare the integrals when the variables don't refer to the same particle anymore?

If labels 1 and 2 are integration arguments, can I write it as ##x_1## and ##x_2##? Which means that ##x_1## and ##x_2## are integration variables? And I can write the integrals as $$\int{\phi*(x_1,x_2)\psi(x_2,x_1) dx_1dx_2}$$ and $$\int{\phi*(x_2,x_1)\psi(x_1,x_2) dx_1dx_2}$$ But, I still...

Homework Statement [/B] Let P be the exchange operator: Pψ(1,2) = ψ(2,1) How can I prove that the exchange operator is hermitian? I want to prove that <φ|Pψ> = <Pφ|ψ>Homework Equations [/B] <φ|Pψ> = <Pφ|ψ> must be true if the operator is hermitian. The Attempt at a Solution [/B] <φ(1,2) |...