Uniqueness of eigenvector representation in a complete set of compatible observables

Sakurai states that if we have a complete, maximal set of compatible observables, say A,B,C… Then, an eigenvector represented by |a,b,c….>, where a,b,c… are respective eigenvalues, is unique. Why is it so? Why can’t there be two eigenvectors with same eigenvalues for each observable? Does maximality of the set has some role to play in … Read more

Entanglement and the double slit experiment

Is the double slit experiment an example of entanglement when it seems as if the photon is going through both slits? Or put another way, is it at this stage when we attempt measurement we see a photon on one side affect the photon on the other side? Do entangled particles have to be made … Read more

How to apply an algebraic operator expression to a ket found in Dirac’s QM book?

I’ve been trying to learn quantum mechanics from a formal point of view, so I picked up Dirac’s book. In the fourth edition, 33rd page, starting from this:ξ|ξ′⟩=ξ′|ξ′⟩ (Where ξ is a linear operator and all the other ξ′‘s are eigen-(value|ket)s.) ,and this:ϕ(ξ)=a1ξn+a2ξn−1⋯an=0 (where ϕ is an algebraic expression) He has deduced ϕ(ξ)|ξ′⟩=ϕ(ξ′)|ξ′⟩ I understand … Read more

Vector space of C4\mathbb{C}^4 and its basis, the Pauli matrices

How do I write an arbitrary 2×2 matrix as a linear combination of the three Pauli Matrices and the 2×2 unit matrix? Any example for the same might help ? Answer A slow construction would go… (abcd)=a(1000)+b(0100)+c(0010)+d(0001) (1000)=12(1001)+12(100−1)=1212+12σ3 (0100)= … ⟹(abcd)=a212+a2σ3+ … (other combintations of the four matrices) AttributionSource : Link , Question Author : chakra , Answer … Read more

Bohr model with all Quantum numbers for hydrogen atom

bohr model: En=−Rn2(1+memp) can we developed bohr model with all Quantum Numbers of the Hydrogen Atom? R(r) Principal quantum number n=1,2,3,4,5,…,n P(θ) Orbital quantum number ℓ=n−1 F(φ) Magnetic quantum number mℓ=−ℓ,ℓ+1,0,ℓ,ℓ−1… Spin quantum number ms=+12,−12 Q numbers Answer The Bohr model was extended by Arnold Sommerfeld. Sommerfeld was able to predict all the Hydrogen atom … Read more

Existence of creation and annihilation operators

In a multiple particle Hilbert space (any space of any multi-particle system), is it sufficient to define creation and annihilation operators by their action (e.g. mapping an n-particle state to an n+1-particle state) or does one have to do anything else, like “proving existence”. Intuitively, one can apriori say it exists if you can write … Read more

Is there a simple approximation to calculate the index of refraction of water?

A very rough approximation from first principles, from the elementary charge and hbar, would suffice. But is there such an approximation at all? (Alternatively, if water is too difficult: is there any other material or gas for which such a calculation is possible?) Answer You can definitely do a calculation for Helium gas under standard … Read more

Conservation Laws and Symmetries

Usually, in Quantum Mechanics, an observable is an operator on the space of the possible quantum states (labelled as |ψ⟩). If this quantity is conserved, in the meaning that the associated operator ˆD is constant: dˆDdt=0 it has to commute with the Hamiltonian operator ˆH. To prove this, in the Heisenberg picture of Classical QM, … Read more

What is the Hubbard-Holstein model?

Please explain as simply as possible what the Hubbard-Holtstein model is and what it is used for. Answer The Hubbard-Holstein model is a electron-phonon model with Hamiltonian H = −t∑iδσc†iσci+δσ+U∑ini↑ni↓+ω0∑ib†ibi+g∑iσniσ(b†i+bi). Quoting R. Ramakumar, A. N. Das, Polaron cross-overs and d-wave superconductivity in Hubbard-Holstein model, arXiv:cond-mat/0611355, Here t (>0) is the hopping energy between molecules at lattice site … Read more