UCSD Physics Problems Worksheet

User Generated

Fhcxvqf6

Science

University of California San Diego

Description

finish question number 10,11,12,14,15

show work

....................................................

Unformatted Attachment Preview

Quantum Physics 2D Key Homework Ch. 6,8,9 Oct., 2018 These problems are to be worked on in discussion. They contain most of the physics concepts needed to solve other problems. The other problems have solutions that can be used to help you with these more basic problems. The solutions to these problems are intentionally not given so you can learn to really solve things yourself. For each problem, state the physics principle (in words) that you will use to solve the problem and choose the best formulas from the formula sheet to answer the problem. Then write an algebraic expression for the answer based on the formulas and finally compute a numerical result if that is possible The text in blue is provided to help you get started on the problem. Don’t expect similar text on the quiz. Do be ready to find the formulas needed on the formula sheet which is on Tritoned and will be provided with the quiz. 1. In 1923, De Broglie realized that to maintain Lorentz invariance, the wave function for a free particle should depend on the Lorentz scalar pµ xµ . Schrödinger postulated that the wave function should be complex, implying that ψf ree−particle = Ceipµ xµ /h̄ where C is just a normalization constant. a) From this formula, derive the frequency of this free particle. Hint: The definition of the frequency is the number of oscillations per second at a fixed location. Using your relativity dot product, write the de Broglie’s wave moving in the x direction, then set the location to x = 0. Explain why the frequency appears like cos(2πf t) in a wave using the definition of the frequency. Now just see what corresponds to the frequency in the de Broglie wave. b) Similarly, derive the wavelength (Length in x of a full oscillation at a fixed time) for the de Broglie wave. c) We always use the kinetic energy to give the frequency. How would physics change if we used the total energy E including the rest mass? (Hint: Factor the energy exponential and consider the relative phases of electrons.) d) Plug this free particle wavefunction into the 1D Schrödinger equation to find out what that equation gives for a free particle. e) In relativistic QM, ψf ree−particle = Ce−ipµ xµ /h̄ is also a solution to the Dirac equation, also with momentum +p but going backward in time. Can you imagine what this means? This is really hard but interesting to think about. 2 2. A 1D free particle (no forces) has a wavefunction ψf ree (x, t) = √ 1 ei(px−Et)/h̄ 2πh̄ where p and E are constants. a) Show that Eop = ih̄ ∂∂t is the total Energy operator because Eop ψf ree (x, t) = Eψ(x, t). These operators are derived in the lecture notes. b) Find the differential operator for the momentum. c) Find the NR differential operator for the Kinetic energy. d) This free particle wavefunction is said to be an eigenstate of momentum and of energy, or a definite 1 momentum state. Consider the state ψ(x, t) = √2πh̄ cos((px − Et)/h̄). What momenta are present in that state? Hint: Try writing this as a linear combination of definite momentum states. 12 )x e) A free electron has a wavefunction ψ(x) = ei(3.5×10 kinetic energy in electron volts? where x is measured in meters. What is the electron’s 3. It can be proven that if the operators for two physical variables don’t commute, then there is an uncertainty principle between those two physical variables. The most basic uncertainty principle is between a coordinate and its “conjugate” momentum. For example, since the laws of physics are invariant under translations in the x-direction, momentum in the x-direction is conserved. Momentum in the x-direction, px , is the “conjugate” of the coordinate x. You have shown above that the operator for the physical variable px is ∂ . The operator for the physical variable x is just x(op) = x (when wave functions are written in terms = h̄i ∂x p(op) x of the coordinate x). and x(op) , a) Compute the commutator between p(op) x h i h (op) (op) p(op) ≡ p(op) − x(op) p(op) x ,x x x x i by carefully computing h i h i (op) (op) p(op) − x(op) p(op) ψ(x) ≡ p(op) ψ(x). x x x x ,x Since this is your first time computing a commutator with differential operators, it is wise to keep the function ψ(x) on the right to avoid dropping important terms containing the derivative. When you are done, drop the wavefunction on the right to quote the correct expression for the commutator. h i (op) . b) Now compute the commutator p(op) y ,x c) The uncertainty principle can be written as (∆p)2 (∆x)2 ≥ h 2i [p(op) , x(op) ]i2 . Show that your result is consistent with this expression. This works for any pair of operators. 4. The nuclear potential that binds protons and neutrons in the nucleus of an atom is often approximated by a square well. Imagine a particle confined in a 1D infinite square well (a box) of length 10−5 nm (a typical nuclear diameter). a) Calculate the energy of the photon that is emitted when a proton undergoes a transition from the 2nd excited state to the ground state. b) Calculate the energy of an electron in the 2nd excited state of the box. 2 5. In a region of space, a particle with zero energy has a wavefunction: ψ(x) = Axe−ax . What is the potential energy function V (x). Hint: Use thge Schrödinger equation to find V (x). 3 6. A 1D Harmonic Oscillator with mass m and classical angular frequency ω is in a linear combination of energy eigenstates. At t = 0, it is in the state ψ(x, 0) = √12 (u0 (x) + iu1 (x)), where un is the nth energy eigenstate of the HO. a) What is the wavefunction at a later time t ? Recall that the time dependence of energy eigenstates is quite simple, ψn (x, t) = un (x)e−iEn t/h̄ . b) What is the expected value of x as a function of time for the above state? This problem is not difficult but requires the use of several important ”Algebra of Quantum Mechanics” skills which I review here in blue. First, the particle is in the state ψ, the expectation value of x (for example) is expressed in Dirac Notation as hψ|x(op) |ψi. Computationally, this expression means: hψ|x (op) Z∞ |ψi ≡ ψ ∗ x(op) ψdx −∞ When you plug in the time dependent ψ be careful to take the complex conjugate when you should. When we operate on an energy eigenstate of the Harmonic Oscillator with x(op) or p(op) , we can write the result again in terms of the energy eigenstates. We have shown by example in lecture (and it can be proven) that s xun =  √ h̄ √ n + 1un+1 + nun−1 2mω s  √ mh̄ω √ n + 1un+1 − nun−1 . 2 This allows us to rather easily compute the expected value of x or p in any HO state. Finally remember that the energy eigenstates are orthonormal, making all of the integral essentially trivial to perform. pun = i hui |uj i = δij c) What is the expected value of p as a function of time? d) At t = 0, a Harmonic Oscillator with mass m and classical angular frequency ω is in the state ψ(x, 0) = √ (u2 (x) − u3 (x))/ 2. What is the expected value of x as a function of time? 7. What is the expected value of x2 in the ground state of a harmonic oscillator? Hint: Use the normal formula for expected value and apply the x operator twice. 8. A particle of mass m moves in a three-dimensional box with sides L . It is found that the ground state has an energy of 3 eV, that the first excited state has an energy of 6 eV, and that the second excited state has an energy of 9 eV. Recall that the 3D box is separable in Cartesian coordinates. When a problems separates like this, the total energy is the sum of energies from each coordinate and the wavefunction is the product of wavefunctions from the three coordinates. This product means that the state in x is the one given by the x quantum number AND the wavefunction in y is... a) What are the quantum numbers of the ground state? 4 b) What are (all the possible sets of) quantum numbers of the first excited state? c) What are (all the possible sets of) quantum numbers of the second excited state? d) Give one set of possible quantum numbers of the third excited state e) Write the wavefunction for this third excited state? 9. In any time independent Schrödinger equation problem the time independence leads to the Energy being conserved and the states being Energy eigenstates. To solve the Hydrogen atom problem, we rely on an important additional symmetry to reduce the number of coordinates in the differential equation from 3 to one radial coordinate. a) Describe the important symmetry we use to help solve the Hydrogen problem. b) In a problem without electron spin, there are 4 operators that commute with the Hamiltonian giving rise to conserved quantities. What are they? c) Because these 4 operators don’t all commute with each other, we choose two mutually commuting operators in addition to the Hamiltonian on which to base our solution. Which pairs of operators have zero commutator? d) Write out the 2s wavefunction of Hydrogen in terms of (r, θ, φ). e) Write out the 2p(m=0), ψ210 , wavefunction of Hydrogen. 10. A Hydrogen atom is in the n = 3 state. Recall that an energy eigenstate ψn`m of Hydrogen has 3 quantum numbers if we don’t consider spin. The simple Energy formula of Rydberg only depends on n and the corrections to Rydberg are very small. Angular momentum is quantized and measurements can only return one of the eigenvalues of the operator. Given just n = 3 we have not specified the angular quantum numbers but with 4d we have specified ` = 2. a) If a measurement of L2 is made, what are the possible outcomes? b) If a measurement of Lz is made, what are the possible outcomes? c) If a measurement of Ly is made, what are the possible outcomes? d) If a measurement of S 2 , the square of the electron’s spin is made, what are the possible outcomes? Why do we not bother to label any Hydrogen states with a quantum number giving the electron’s spin? e) For an electron in the 4d shell, what is the expected value of L2 ? 11. Many calculations of important atomic physics problems come down to expectation values (or similar matrix elements) of some power of the radius in Hydrogen states. It turns out that these calculations are rather easy, so we will learn to do them. We use the orthonormality of the Spherical Harmonics Y`m , and a simple formula for a definite integral over the radial coordinate that comes up all the time in these calculations with Hydrogenic wavefunctions. This formula comes from integration by parts but since it is used so often, it is important for us to remember it. Z ∗ Y`m Y`0 m0 dΩ = Z∞ 0 Z ∗ Y`m Y`0 m0 d cos θ dφ = δ``0 δmm0 dr rn e−ar = n! an+1 5 ψn`m (r, θ, φ) = Rn` (r)Y`m (θ, φ) Just use the orthonormality to do the angular integral, then do the radial integral. a) What is the expected value of the radius squared, hψ21m |r2 |ψ21m i, for the 2p state of hydrogen? b) What is the expected value of the radius, for the ground state of hydrogen? How does this compare to the Bohr radius? c) What is the expected value of 1r , for the ground state of hydrogen? 12. The Hydrogen states can be written in terms of their radial wavefunctions and the spherical harmonics. a) Write out the 2p state ψ211 of Hydrogen. b) Write out the 2s state ψ200 of Hydrogen. c) Why does the 2p state go to zero at r = 0? Why doesn’t the 2s state? d) Which state had more ”nodes”, that is more zeros of the wavefunction excluding those at infinity and r = 0. 13. Electrons with orbital angular momentum are very similar to a current loop which generates a magnetic dipole e ~ field. The calculated magnetic dipole moment is proportional to the angular momentum: µ ~ = 2m L. When a magnetic field is applied, these magnetic moments have an energy that depends on thier orientation relative to ~ Choosing the field to be along the z direction, we have: the field: E = −~µ · B. ~ = E = −~µ · B e eh̄ e Lz B = m` h̄B = m` B = µB Bm` 2me 2me 2me The Zeeman Effect is the splitting of atomic states in a magnetic field. a) An atom has a single valence electron in the 2p state. Since all the other electrons add up to have net zero orbital angular momentum, the magnetic moment is entirely due to this valence electron. What energy shifts do we expect to see in a 0.5 Tesla magnetic field? What element would have this single 2p valence electron? b) What Energy shifts would we see for a single valence electron in 3d state? What element would have this single 3d electron. c) Why do we expect the magnetic field to split the state into an odd number of energies? 14. The expected Zeeman effect was not really observed in atoms. Rather than an odd number of lines with equal spacing as expected for orbital angular momentum, an even number of lines was usually observed and the splitting between them was not the same. This has been called the Anomalous Zeeman Effect because it was very hard to understand. Stern and Gerlach separated a beam of Silver atoms into two beams in a B field with a large gradient. Silver has a single valence electron in the 5s state which would not split according to the normal Zeeman Effect. The Silver beam has a magnetic moment because the electron has internal angular momentum that we call spin. Spin one-half electrons have a magnetic moment of one Bohr Magneton, the same size as the magnetic moment due to ` = 1 orbital angular momentum. That is, they have a gyromagnetic ratio of g = 2. ~ The magnetic moment operator is These magnetic moments give an energy in a magnetic field of E = −~µ · B. ~ and the resulting energies are thus quantized. Choosing the field to be along the z direction, proportional to S we have: ~ = ge Sz B = ge ms h̄B = g eh̄ ms B = µB Bgms E = −~µ · B 2me 2me 2me 6 ~ must be added to those due to S. ~ Addition For electrons with orbital angular momentum, the splitting due to L ~ ~ ~ of angular momentum J = L + S, helps but because the gyromagnetic ratio for spin is 2 while the gyromagnetic ratio for orbital is 1, its still complicated. a) Consider the original Stern-Gerlach experiment in which the magnetic moment of Silver atoms is entirely due to the spin of a single valence electron. What is the difference in energy between the two spin states of the Silver atom in a 1 T magnetic field? b) An electron is placed in a 2T magnetic field. The magnetic moment of the electron is entirely due to its spin. What frequency of EM radiation will cause transitions between the two spin states? 15. An electron is in the + h̄2 eigenstate of Sx at t = 0. That is its spin is up along the x direction. The electron is in a B-field Bz in the z direction. The state will be given as a linear combination of spin-up and spin-down (along   a the z direction). This can be expressed as a 2-component vector χ = where a is the amplitude to have b spin-up and b is the amplitude to have spin-down. The state should be normalized to 1 so that |a|2 + |b|2 = 1.   0 1 Lets determine what that state is. The Sx operator is given by Sx = h̄2 . We want Sx χ = + h̄2 χ to be 1 0 q           1 0 1 a a b a 2 h̄ h̄  q to keep χ in that eigenstate. So we have 2 or = 2 = , implying χ = 1 1 0 b b a b 2 normalized. a) What is the electron’s state as a function of time? b) What is the expected value of Sx as a function of time? The expected value is computed in this vector-state, matrix-operator notation as hSx i = χT ∗ Sx χ. c) What is the expected value of Sy = h̄ 2  0 −i i 0  as a function of time? 16. Ten identical, non-interacting, spin 21 particles with mass m are placed into a cubical box of side L = 0.25 nm. What is the ground state energy of this system? Hint: Because of the two spin states, we can put 2 electrons into each spatial state of the cubic box. The spin state of the 2 electrons will be the antisymmetric state. 17. A Hydrogen atom is in the 3d state. If a measurement of J 2 is made, what are the possible outcomes? In the calculation of the Hydrogen Fine Structure, we find that the energy depends on the total angular momentum quantum number j and not on ` or s. The formulas in the Atomic physics section should be used. The limits on j are given and the eigenvalues of J 2 are given in terms of j.What are the Fine-Structure corrections to the energy for these two states? 18. Every electron in the universe is the same as any other electron. It is a symmetry of physics, that if you interchange any pair of electrons, the Schrödinger equation is invariant. Consider one pair of electrons which we call electron-1 and electron-2. Define the interchange operator P12 to simple interchange the two electrons. Hψ1 ψ2 = Eψ1 ψ2 HP12 ψ1 ψ2 = EP12 ψ1 ψ2 7 a) Use the above equations to show that the P12 operator commutes with the Hamiltonian. b) Use that fact that if we operate twice with P12 we get back to the same state, to show what the eigenvalues of the P12 operator are. c) The overall state of two electrons must be antisymmetric under interchange. Usually, we can factorize the spin state of two electrons from the spatial state: ψ = ψspin ψspace . Since the overall state of two electrons has to be antisymmetric under interchange, we can have a symmetric spin state with an antisymmetric space state, or vice versa. Lets delve into the spin states of two electrons. Its very common to use arrows to designate whether the spin of an electron is up or down since these are the only two states of an electron and to use the first arrow for electron-1 and the second for electron-2. So, for example, the state ↑↓ says that electron-1 has spin up and electron two has spin down. Since there are only 2 spin states for any electron, there are just 4 states of two electrons: ↑↑, ↑↓, ↓↑, and ↓↓. Of these four states, which are symmetric under interchange and which are antisymmetric. d) From the two states that are neither symmetric or antisymmetric, make one linear combination that is antisymmetric and one that is symmetric. Multiply by a constant to normalize these states to one pair of electrons. e) Write the properly-normalized, symmetric-under-interchange spin state(s) of two electrons? f) What is the properly-normalized, antisymmetric-under-interchange spin state of two electrons? 19. The Helium ground state has two electrons in the 1s state and in the antisymmetric spin state. The Helium first excited state has one electron in the 1s state and the other electron in the 2s state. Recall that the big effect on the energies of the states is the Coulomb repulsion between the two electrons. We have found that the spatial antisymmetric state has significantly lower energy than the symmetric one because of this repulsion. a) Which of the possible (1s)(2s) states will have the lowest energy? b) What is the symmetry of the spin state of the two electrons? c) Write an overall antisymmetric state that is (one of) the degenerate first excited state(s) of Helium. 20. In Hydrogen, the energies of states depend on n and j but not on `, even with the fine structure corrections included. This degeneracy in ` is strongly broken in atoms with multiple electrons. a) What is the explanation for the fact that states with higher ` have higher energy in atoms? b) The order of states as the periodic table is filled up is still unclear with this big change that lower-` states have lower energy. We can get the order of orbitals right for the atoms we find in nature by following two simple rules. First pick the state with the lowest n + ` and if that is equal, pick the state with the lowest n. With this, write down the order of n` orbitals in atoms up to Rutherfordium (Z=104). (Not for each atom, just 1s, 2s, 2p, ...). c) Finally, in choosing the ground state of an atom, we should avoid ”paired electrons”. That is avoid two electrons in the same spatial state that therefore must be in the symmetric spatial state and have maximum Coulomb repulsion. This gives rise to the Chemist’s rather crude chart that shows that we put one spin up electron ↑ into each m` state until they are all taken, then start pairing the electrons designated by ↑↓. Draw the Charts for Nitrogen and for Iron. d) With all the Coulomb repulsion between electrons, it seems surprising that the atomic shell model works well, particularly for large Z. Why does the atomic shell model work well?
Purchase answer to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Explanation & Answer

Hi! this is the solution. Do not hesitate to write to me if you need anything else. 😎

Scanned by...


Anonymous
Just what I was looking for! Super helpful.

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4

Related Tags