Write down the complete wave function for the hydrogen atom when the electron's quantum numbers are n = 2, l=1, ml=-1 Write down the normalization condition(s). Verify that the wave function is normalized. Useful Integrals: sine de cos+cos(38) rexp(-ur)dr=
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Write down the complete wave function for the hydrogen atom when the electron's quantum numbers are n = 2, l=1, ml=-1 Write down the normalization condition(s). Verify that the wave function is normalized. Useful Integrals: sine de cos+cos(38) rexp(-ur)dr=
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- Write down the confi gurations for the ground states of calcium and aluminum. What are the LS coupling quantum numbers for the outside subshell electrons? Write the spectroscopic symbol for each atom1.5 The time-averaged potential of a neutral hydrogen atom is given by - ar Ф 4 TTE0 ar 1 + 2 r where q is the magnitude of the electronic charge, and al = Bohr radius. Find the distribution of charge (both continuous and discrete) that will give this potential and interpret your result physically. ao/2, ao being theThe Lennard-Jones parameters for argon are: A = 1.022 x 10-7 J m° and B = 1.579 x 10-134 J m12. Calculate the distance at which the energy will be minimum and calculate the minimum energy. Draw the energy profile also. I.
- Consider the hydrogen atom. (a) For the ground state (n = 1), first excited states (n = 2), and second excited states (n = 3): 1. Calculate the energy En in eV. 2. Calculate all possible values of the angular momentum L and z-component angular momentum Lz (you can leave your values in terms of ¯h – e.g., L = 2¯h). 3. For each value of n and L, sketch the radial wave function and radial probability density. (b) Prove that, in the ground state, the electron is most likely to be found at the Bohr radius aB.A Construct the wavefunction V(r, 0, ¢) for an H atoms' electron in the state 2pz. Please note that in order to have a real-valued wavefunction of pr orbital(see below), you need to do a linear superposition of the corresponding spherical harmonics for the angular part. Use the spherical harmonics table below. Show that the superposition you selected indeed results in a real orbital; however, you do not need to simplify the expressions further or normalize the wavefunction. Px 1/2 Yg = ()"" (5 cos 0 -3 cos 0) cos e %3D (4x 21 12 64л/ 1/2 sin e (5 cos? e- 1)eti 87 -y Y = (3 cos²0 – 1) 105 1/2 32 sin e cos de2ie (167 15 12 sin e cos betie 35 12 (647 sin de i B Now consider an excited state of He atom with electron configuration 1s 2s'. In general, the wavefunction is a state: V(r, 0, 0, 02) = V(r,0, ø)V.. where V(r, 0, 6) and V,, represent the spatial and the spin part. The spatial part is constructed from the wavefunctions of the 1s' and 2s' orbitals denoted as o (r, 0, ø) and o (r, 0,…solve the problem An electron with angular momentum {= 1 exists in the state X = A Where A is the normalization constant. A) Find the value of A B) If a measurement of Ldis made, what values will be obtained, and with what probabilities?
- Is the total system energy of a dipole a positive potential or a negative potential? Explain your reasoning. HINT: consider the total energy of a bound state atom, where there is a positive nucleus and an electron in orbit/orbitals. Is the electron energy positive or negative energy?a) Calculate the energy of the emissive transition with the lowest energy possible for the Lyman series, for a mole of hydrogen atoms. Express your answer in joules/mol. b) Is this transition in the visible spectral domain? If not, in which region is it located?The wave function for H-atom in 1s state is given below. Explain the radial probability density of H-atom in 1s state and also write an expression for the same. Sketch schematically the plot of this vs. radial distance.
- calculate the expectation value for the potential energy of the H atom with the electron in the 1s orbital. Compare your result with the total energy. What is the kinetic energy of H atom in this state? Verify the virial theorem for the Coulomb potential.The three lowest energy levels of a hydrogen atom are -13.6 eV, -3.4 eV, and -1.5 eV. Assume that there is only one way to occupy any one of these levels. Calculate the relative probability that a hydrogen atom in thermal equilibrium in a star, at temperature T = 9674 K, is in its first excited state (at -3.4 eV) relative to its ground state (at -13.6 eV). Write your answer in exponential form. Recall that Boltzmann's constant can be written as 8.617 x 10-5 eV K-1.Write down the wave function of an electron in a ψ(n=7, l=3, m=−1) state of a hydrogen atom. What is the expectation value ⟨E⟩ for the total energy?