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Consider the hydrogen atom and its eigenstates, omitting any effects of fine structure (spin- orb...
In the Bohr model of the hydrogen atom, the allowed orbits of the electron (labeled n...
In the Bohr model of the hydrogen atom, the allowed orbits of the electron (labeled n = 1, 2, 3, …) have angular momentum , orbital radii , and energies . In these expressions me is the mass of the electron. In an exotic atom the electron is replaced by a different subatomic particle that has the same charge as an electron but a different mass. Two examples that have been studied are muonic hydrogen, in which the electron is...
In the Bohr model of the hydrogen atom, the allowed orbits of the electron (labeled n...
In the Bohr model of the hydrogen atom, the allowed orbits of the electron (labeled n = 1, 2, 3, …) have angular momentum , orbital radii , and energies . In these expressions me is the mass of the electron. In an exotic atom the electron is replaced by a different subatomic particle that has the same charge as an electron but a different mass. Two examples that have been studied are muonic hydrogen, in which the electron is...
Consider the real energy levels of hydrogen which include the fine structure (relativistic spin orbit coupling)...
Consider the real energy levels of hydrogen which include the fine structure (relativistic spin orbit coupling) (a) If the system is placed in a strong magnetic field of about 40000 Gauss, how many levels does the n=3 state splits into? (b) What is the energy difference (in eV) between them? The energy levels of the hydrogen atom including the fine structure correction (but excluding Zeeman effect corrections) can be written as
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the...
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the subscript of E is n, the principal quantum number. The other two numbers are the 1 and m values, find the expectation values of H (you may use the eigenvalue equation to evaluate for H), L-(total angular momentum operator square), Lz (the z-component of the angular momentum operator) and P (parity operator). Draw schematic pictures of 1 and...
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic...
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic fieldB given by: v x (-V(r)) v x E B=- where (r) is the electric potential. This magnetic field interacts with the magnetic moment u of the electron given by -S, =n me where S is the electron spin. Assuming non-relativistic mechanics, show that the Hamiltonian representing this effect (spin-orbit coupling) for a spherically-symmetric electric potential is: 1 dφ(r) S.L ΔΗ [6] r dr...
PLEASE COMPLETE B) and stay tuned for my following 2 questions where I will ask part c) and d). P...
PLEASE COMPLETE B) and stay tuned for my
following 2 questions where I will ask part c) and d). Part a) has
already been posted.
The lowest energy state of a hydrogen-like atom has total angular momentum J-1/2 (from the l-O orbital angular momentum and the electron spin s 1/2). Furthermore, the nucleus also has a spin, conventionally labeled I (for hydrogen, this is the proton spin, 1 1/2). This spin leads to an additional degeneracy. For example, in the...
4. (20 points). Consider a deuterium atom (composed of a nucleus with spin I - 1...
4. (20 points). Consider a deuterium atom (composed of a nucleus with spin I - 1 and an electron) The electronic angular momentum is J- L S, where L is the orbital angular momentum of the electron and S is its spin. The total angular momentum of the atom is F-J+I. The eigenvalues of J2 and F2 are j(j + 1)n° and f(f+1)ћ, respectively a. What are the possible values of the quantum numbers j and f for a deuterium...
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value...
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value of the Hamiltonian in the (sa)'(2a)' excited state of helium is given by E = $42.0) (Avo ) anordes ++f63,(-) (%13-12 r) 62(e)drz + løn.(r.) per 142, (ra)]" drų dr2 - / 01.(ru) . (ra) Anemia 02.(r.)61.(r.)dr; dr2 (1) Use the approximate, antisymmetrized triplet state wave function for the (Isa)'(280)' state as discussed in class. Hint: make use of the orthonormality of the hydrogenic...
The function ψ2px-1(ψ2,1,1+ψ2,1-1) describes an electron in the 2px state of a hydrogen-like atom (with unspecified...
The function ψ2px-1(ψ2,1,1+ψ2,1-1) describes an electron in the 2px state of a hydrogen-like atom (with unspecified spin). Functions ψη..my are normalized egenfuntions of the energy operator (A), the square of angular momentum operator (12), and the z-component of angular momentum operator (Lz), that is 4. E1 a) Show that the function ψ2px is an eigen function of both the energy operator and the square of angular momentum operator. Find the corresponding eigenvalues. b) Determine the expected value and the uncertainty...
Consider a hydrogen atom in its ground state. The hyperfine interaction between the magnetic moment of...
Consider a hydrogen atom in its ground state. The hyperfine interaction between the magnetic moment of the proton and the magnetic moment of the electron is described by the Hamiltonian: H =A S.S, where S, is the spin of the electron, S, is the spin of the proton, and A is a constant. The ground state is split by the hyperfine coupling. Obtain the energies of the split levels.
Consider the real energy levels of hydrogen which include the fine structure (relativistic spin orbit coupling) (a) If the system is placed in a strong magnetic field of about 40000 Gauss, how many levels does the n=3 state splits into? (b) What is the energy difference (in eV) between them? The energy levels of the hydrogen atom including the fine structure correction (but excluding Zeeman effect corrections) can be written as
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the subscript of E is n, the principal quantum number. The other two numbers are the 1 and m values, find the expectation values of H (you may use the eigenvalue equation to evaluate for H), L-(total angular momentum operator square), Lz (the z-component of the angular momentum operator) and P (parity operator). Draw schematic pictures of 1 and...
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic fieldB given by: v x (-V(r)) v x E B=- where (r) is the electric potential. This magnetic field interacts with the magnetic moment u of the electron given by -S, =n me where S is the electron spin. Assuming non-relativistic mechanics, show that the Hamiltonian representing this effect (spin-orbit coupling) for a spherically-symmetric electric potential is: 1 dφ(r) S.L ΔΗ [6] r dr...
PLEASE COMPLETE B) and stay tuned for my
following 2 questions where I will ask part c) and d). Part a) has
already been posted.
The lowest energy state of a hydrogen-like atom has total angular momentum J-1/2 (from the l-O orbital angular momentum and the electron spin s 1/2). Furthermore, the nucleus also has a spin, conventionally labeled I (for hydrogen, this is the proton spin, 1 1/2). This spin leads to an additional degeneracy. For example, in the...
4. (20 points). Consider a deuterium atom (composed of a nucleus with spin I - 1 and an electron) The electronic angular momentum is J- L S, where L is the orbital angular momentum of the electron and S is its spin. The total angular momentum of the atom is F-J+I. The eigenvalues of J2 and F2 are j(j + 1)n° and f(f+1)ћ, respectively a. What are the possible values of the quantum numbers j and f for a deuterium...
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value of the Hamiltonian in the (sa)'(2a)' excited state of helium is given by E = $42.0) (Avo ) anordes ++f63,(-) (%13-12 r) 62(e)drz + løn.(r.) per 142, (ra)]" drų dr2 - / 01.(ru) . (ra) Anemia 02.(r.)61.(r.)dr; dr2 (1) Use the approximate, antisymmetrized triplet state wave function for the (Isa)'(280)' state as discussed in class. Hint: make use of the orthonormality of the hydrogenic...
The function ψ2px-1(ψ2,1,1+ψ2,1-1) describes an electron in the 2px state of a hydrogen-like atom (with unspecified spin). Functions ψη..my are normalized egenfuntions of the energy operator (A), the square of angular momentum operator (12), and the z-component of angular momentum operator (Lz), that is 4. E1 a) Show that the function ψ2px is an eigen function of both the energy operator and the square of angular momentum operator. Find the corresponding eigenvalues. b) Determine the expected value and the uncertainty...
Consider a hydrogen atom in its ground state. The hyperfine interaction between the magnetic moment of the proton and the magnetic moment of the electron is described by the Hamiltonian: H =A S.S, where S, is the spin of the electron, S, is the spin of the proton, and A is a constant. The ground state is split by the hyperfine coupling. Obtain the energies of the split levels.
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