You have 3 spin-1/2 particles in the following state: |ψ⟩ = 1/√2 (|+z⟩1 |+z⟩2 |+z⟩3 + |-z⟩1 |-z⟩2 |-z⟩3 ) Write the state with particle 1 and particle 2 in the |±y⟩ basis, and particle 3 in the |±x⟩ basis
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You have 3 spin-1/2 particles in the following state:
|ψ⟩ = 1/√2 (|+z⟩1 |+z⟩2 |+z⟩3 + |-z⟩1 |-z⟩2 |-z⟩3 )
- Write the state with particle 1 and particle 2 in the |±y⟩ basis, and particle 3 in the |±x⟩ basis
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- Recall for an the hydrogenic (single electron) atom 2s (r) = 2,0,0 (r, 0, 4) Φ2p (r) = Φ2,1,0 (r,θ, φ) - = 2p (7) = 2p_ (F) = 2,1,1 (r, 0, 6): = 2,1,-1 (r, 0,6) 1 4√2π/² p 1 3/2 ao 4√/2πа = 2 δεν παρ Tº 3/2 ao 8√πа 3/2 ao 1) e-r/2² ao e ○ (02s (71)2p, (72) + O2p. (71)02s (72)) O 02s (1) 2po (2) ○(28 (71)2p, (72) – $2p. (71)¢2s (72)) O 02s (1)02s (F2) T -T 12a0 •/200 cos 0, /2ao sin 0 etic. r/2ao sin 0 e-iç Consider the helium atom (two electron system). Suppose the spin part is one of the triplet. Which of the following can be a possible space part?(a)How many sets of quantum numbers are possible for a hydrogen atom for n = 4,? ). (b) Write out a set of possible values for the quantum numbers n, ℓ, mℓ, and ms for each electron if all states are occupied including n=4. (c)Write table of occupancy of quantum numbers: n, ℓ, mℓ, and ms for Arsenic As, including spin orientations.A spin-particle is in the spin state |A), described by the ket 7 i |4) = 5√2 tu) +5√2 tu). (a) Verify that A) is normalised. (b) Using the spinor representation (+₂) = ₁ | +) = √/2₁ 11³) = ₁ 1 +2) = [8] find the values of c₁ and c₂ for which |A) = C₁|1₂) + C₂l+₂). (c) If the observable S₂ is measured in the spin state |A), what values can be obtained and what are their probabilities? (d) Find the expectation value of S₂ in the spin state [A). (e) With reference to the properties of angular momentum, explain briefly how the results of the Stern-Gerlach experiment provide evidence of the existence of spin.
- Calculate the number of angles that L can make with the z-axis for an l=3 electron.(d) The following orbital belongs to the 3d subshell of the Hydrogen atom: r Y(r, 0, 0) = A(Z) θ, φ) 2 r e 3ao sin² (0) e²i зао where A and ao are constants. Using the operator for the z-component of orbital angular momentum (L₂ = -ih d/do) determine the m, for this particular orbital. (e) Consider the wavefunction, r r Y(r,0,0) = A-e 2do cos(0) do (i) Identify the radial part of this orbital function and the number of radial nodes. (ii) Identify the angular part of the orbital function and the number of angular nodes. Z (iii) Using this information and the L₂ = -ih d/do operator obtain the n, 1, and, m quantum numbers and identify the orbital.Suppose you measure the angular momentum in the z-direction L, for an /= 2 hydrogen atom in the state | > 2 > |0 > +i/ |2 >. The eigenvalues of %3D V10 10 Lz are – 2h, -ħ, 0, ħ, 2ħfor the eigenvectors | – 2 >, |– 1>, |0 >, |1 >, |2 >, respectively. What is AL,? V31 10 7 19 25
- (a) How many angles can L make with the z -axis for an l = 2 electron? (b) Calculate the value of the smallest angle.How many different possible states can there be for an atom with n =5 and l = 1 ?There exists in nature a particle known as the muon. It is just a heavy electron with a mass mµ = 207me. It decaysin 10−6seconds. Suppose there exists a molecule analogous to H+2(two protons + 1 electron), but with the electronreplaced by a muon:(a) Find the equilibrium separation of the nuclei (R0) in such a molecule.(b) If a rotational state is excited, estimate the wavelength of the emitted radiation in the transition to the groundstate
- In a particular state of the hydrogen atom, the angle between the angular momentum vector L →and the z-axis is u = 26.6°. If this is the smallest angle for this particular value of the orbital quantum number l, what is l?Consider an electron in an external magnetic field in the SzS-direction, $\mathbf{B} B_z \hat{k}S. If the initial spin state of the electron is the eigenstate of SS_x$ with eigenvalue S+\hbar/2$, \begin{enumerate} \item[a)] Find the state of the system at time St$. \item [b)] Show that the system returns to its initial state and calculate the angular frequency. \item [c)] Calculates the expected value of SS_XS, SS_yS and SS_ZS as a function of time. \ end {enumerate}