Questions and Answers of Introduction To Quantum Mechanics

Solve Equation 11.17 to second order in perturbation theory, for the general case Ca(0) = a, Cb(0) = b. Ca Hab h == Habe-icol Cb, i =-=-Hex Cas H'ba eiwot (11.17)
Consider a perturbation to a two-level system with matrix elementswhere τ and α are positive constants with the appropriate units.(a) According to first-order perturbation theory, if the system
The first term in Equation 11.32 comes from the eiωt/2 part of Cos(ωt), and the second from e-iωt. Thus dropping the first term is formally equivalent to writing Ĥ' = (V/2)e-iωt, which is to
You could derive the spontaneous emission rate (Equation 11.63) without the detour through Einstein’s A and B coefficients if you knew the ground state energy density of the electromagnetic field,
As a mechanism for downward transitions, spontaneous emission competes with thermally stimulated emission (stimulated emission for which blackbody radiation is the source). Show that at room
From the commutators of Lz with x, y, and z (Equation 4.122):obtain the selection rule for Δm and Equation 11.76. Sandwich each commutator between (n'ℓ 'm'| and |nℓ m). [L, x]=ihy. [Ly] =-ihx,
The half-life (t1/2) of an excited state is the time it would take for half the atoms in a large sample to make a transition. Find the relation between t1/2 and τ (the “lifetime” of the state).
An electron in the n = 3, ℓ = 0, m = 0 state of hydrogen decays by a sequence of (electric dipole) transitions to the ground state.(a) What decay routes are open to it? Specify them in the
In the photoelectric effect, light can ionize an atom if its energy (ћω) exceeds the binding energy of the electron. Consider the photoelectric effect for the ground state of hydrogen, where the
Obtain the selection rule for Δℓ as follows:(a) Derive the commutation relationUse this, and (in the final step) the fact that r.L = r. (rxP) = 0, to demonstrate thatThe generalization from z to r
A particle of mass m is in the ground state of the infinite square well (Equation 2.22). Suddenly the well expands to twice its original size—the right wall moving from a to 2a—leaving the wave
Check that Equation 11.103 satisfies the time-dependent Schrödinger equation for the Hamiltonian in Equation 11.97. Also confirm Equation 11.105, and show that the sum of the squares of the
Find Berry’s phase for one cycle of the process in Example 11.4. Use Equation 11.105 to determine the total phase change, and subtract off the dynamical part. You’ll need to expand λ (Equation
In Problem 11.1 you showed that the solution to(where k(t) is a function of t) isThis suggests that the solution to the Schrödinger equation (11.1) might beIt doesn’t work, because Ĥ(t) is an
The delta function well (Equation 2.117) supports a single bound state (Equation 2.132). Calculate the geometric phase change when α gradually increases from α1 to α2 . If the increase occurs at a
A particle is in the ground state of the harmonic oscillator with classical frequency ω, when suddenly the spring constant quadruples, so ω' = 2ω, without initially changing the wave function (of
In this problem we develop time-dependent perturbation theory for a multi-level system, starting with the generalization of Equations 11.5 and 11.6:At time t = 0 we turn on a perturbation H' (t), so
For the examples in Problem 11.24 (c) and (d), calculate Cm(t), to first order. Check the normalization condition:and comment on any discrepancy. Suppose you wanted to calculate the probability of
A particle starts out (at time t = 0) in the Nth state of the infinite square well. Now the “floor” of the well rises temporarily (maybe water leaks in, and then drains out again), so that the
A spin-1/2 particle with gyromagnetic ratio γ, at rest in a static magnetic field B0k̂, precesses at the Larmor frequency ω0 = ϒB0 (Example 4.3). Now we turn on a small transverse
A particle of mass m is initially in the ground state of the (one-dimensional) infinite square well. At time t = 0 a “brick” is dropped into the well, so that the potential becomeswhere V0
In this problem we will recover the results Section 11.2.1 directly from the Hamiltonian for a charged particle in an electromagnetic field (Equation 4.188). An electromagnetic wave can be described
We have encountered stimulated emission, (stimulated) absorption, and spontaneous emission. How come there is no such thing as spontaneous absorption?
(a) Prove properties 12.17, 12.18, 12.19, and 12.20.(b) Show that the time evolution of the density operator is governed by the equation(This is the Schrödinger equation, expressed in terms of
(a) Prove properties 12.31, 12.32, 12.33, and 12.34.(b) Show that Tr(ρ2) ≤ 1, and equal to 1 only if ρ represents a pure state.(c) Show that ρ2 = ρ if and only if ρ represents a pure state. pt
Einstein’s Boxes. In an interesting precursor to the EPR paradox, Einstein proposed the following gedanken experiment: Imagine a particle confined to a box (make it a one-dimensional infinite
(a) Show that the most general density matrix for a spin-1/2 particle can be written in terms of three real numbers (α1,α2,α3):where σ1,σ2,σ3 are the three Pauli matrices.(b) In the literature,
Consider the ordinary vectors in three dimensionswith complex components.(a) Does the subset of all vectors with az = 0 constitute a vector space? If so, what is its dimension; if not, why not?(b)
(a) Construct the density matrix for an electron that is either in the state spin up along x (with probability 1/3) or in the state spin down along y (with probability 2/3).(b) Find (Sy) for the
Find the angle (in the sense of Equation A.28) between the vectors a) = (1 + i) î+ (1) ĵ + (i)k and|p) = (4 - i) î+ (0) î+ (2-2i) k.
Given the following two matrices:compute: (a) A+B, (b) AB, (c) [A,B] (d) A̅, (e) A*, (f) A+, (g) det(B)(h) B-1. Check that BB-1 = 1. Does A have an inverse? A = -
Prove Equations A.52, A.53, and A.58. Show that the product of two unitary matrices is unitary. Under what conditions is the product of two hermitian matrices hermitian? Is the sum of two unitary
Using the square matrices in Problem A.8, and the column matricesfind: (a) Aa, (b) a +b(c) a̅Bb(d) ab+. a= 2i 2 b = (1-1) -
Consider the collection of all polynomials (with complex coefficients) of degree < N in x.(a) Does this set constitute a vector space (with the polynomials as “vectors”)? If so, suggest a
By explicit construction of the matrices in question, show that any matrix T can be written(a) As the sum of a symmetric matrix S and an antisymmetric matrix A;(b) As the sum of a real matrix R and
Show that the rows and columns of a unitary matrix constitute orthonormal sets.
Noting that det(T̅) = det(T), show that the determinant of a hermitian matrix is real, the determinant of a unitary matrix has modulus 1 (hence the name), and the determinant of an orthogonal matrix
It’s obvious that the trace of a diagonal matrix is the sum of its eigenvalues, and its determinant is their product (just look at Equation A.79). It follows (from Equations A.65 and A.68) that the
Using the standard basis (î ĵ k̂) for vectors in three dimensions:(a) Construct the matrix representing a rotation through angle θ (counterclockwise, looking down the axis toward the origin)
The 2x2 matrix representing a rotation of the xy plane isShow that (except for certain special angles—what are they?) this matrix has no real eigenvalues. (This reflects the geometrical fact that
In the usual basis (î ĵ k̂), construct the matrix Tx representing a rotation through angle θ about the x axis, and the matrix Ty representing a rotation through angle θ about the y axis.
Show that the first, second, and last coefficients in the characteristic equation (Equation A.73) are:For a 3x3 matrix with elements Tij, what is C1? Cn = (-1)", Cn-1 = (-1)" Tr(T), and Co = det
Show that similarity preserves matrix multiplication (that is, if AeBe = Ce, then AfBf = Cf). Similarity does not, in general, preserve symmetry, reality, or hermiticity; show, however, that if S is
Prove that Tr(T1T2) = Tr(T2T1). It follows immediately that Tr(T1T2T3) = Tr(T2T3T1), but is it the case that Tr(T1T2T3) = Tr (T2T1T3), in general? Prove it, or disprove it.
Show that if two matrices commute in one basis, then they commute in any basis. That is:Use Equation A.64. [T₁, T₂] = 0 ⇒ [T{, T₂] = 0. (A.94)
Consider the matrices(a) Verify that they are diagonalizable and that they commute.(b) Find the eigenvalues and eigenvectors of A and verify that its spectrum is degenerate.(c) Are the eigenvectors
Show that the a̅ computed from Equations A.88 and A.90 are eigenvectors of V. a(1) da(¹) +d₂1a(²) a(2)=d₁a¹)+da(²), = (A.88)
Let(a) Verify that T is hermitian.(b) Find its eigenvalues (note that they are real).(c) Find and normalize the eigenvectors (note that they are orthogonal).(d) Construct the unitary diagonalizing
A hermitian linear transformation must satisfy (α|T̂β) = (T̂α|β) for all vectors |α) and |β). Prove that it is (surprisingly) sufficient that (ϒ|T̂ϒ) = (T̂ϒ|ϒ) for all vectors |ϒ).
Consider the following hermitian matrix:(a) Calculate det (T) and Tr (T).(b) Find the eigenvalues of T. Check that their sum and product are consistent with (a), in the sense of Equation A.93. Write
A unitary transformation is one for which Û+Û = 1.(a) Show that unitary transformations preserve inner products, in the sense that (Ûα|Ûβ) = (α|β), for all vectors |α), β).(b) Show that
Functions of matrices are typically defined by their Taylor series expansions. For example,(a) Find exp (M), if(b) Show that if M is diagonalizable, thenComment: This is actually true even if M is
Prove the following three theorems:(a) For normalizable solutions, the separation constant E must be real. Write E (in Equation 2.7) as E0 + iΓ (with E0 and Γ real), and show that if Equation
Show that E must exceed the minimum value of V(x) , for every normalizable solution to the time-independent Schrödinger equation. What is the classical analog to this statement?if E < Vmin, then
Show that there is no acceptable solution to the (time-independent) Schrödinger equation for the infinite square well with E = 0 or E < 0.
A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states:(a) Normalize Ψ (x,0). (That is, find A. This is very easy, if you exploit
A particle of mass m in the infinite square well (of width a) starts out in the statefor some constant A, so it is (at t = 0) equally likely to be found at any point in the left half of the well.
(a) Construct Ψ2 (x).(b) Sketch Ψ0, Ψ1, and Ψ2.(c) Check the orthogonality of Ψ0, Ψ1 and Ψ2, by explicit integration. If you exploit the even-ness and odd-ness of the functions,
In the ground state of the harmonic oscillator, what is the probability (correct to three significant digits) of finding the particle outside the classically allowed region? Classically, the energy
Use the recursion formula (Equation 2.85) to work out H5 (ξ) and H6 (ξ). Invoke the convention that the coefficient of the highest power of ξ is 2n to fix the overall constant.Equation 2.85 aj +2
This problem is designed to guide you through a “proof” of Plancherel’s theorem, by starting with the theory of ordinary Fourier series on a finite interval, and allowing that interval to
Delta functions live under integral signs, and two expressions (D1(x) and D2 (x)) involving delta functions are said to be equal iffor every (ordinary) function f(x).(a) Show thatwhere c is a real
Check that the bound state of the delta-function well (Equation 2.132) is orthogonal to the scattering states (Equations 2.134 and 2.135). y(x) = √ma e-ma\x\/h²; h E: ma² 2h² (2.132)
What is the Fourier transform of δ(x)? Using Plancherel’s theorem, show thatThis formula gives any respectable mathematician apoplexy. Although the integral is clearly infinite when x = 0, it
In this problem you will show that the number of nodes of the stationary states of a one-dimensional potential always increases with energy. Consider two (real, normalized) solutions (Ψn and Ψm) to
A particle in the infinite square well (Equation 2.22) has the initial wave functionDetermine A, find Ψ (x,t), and calculate (x), as a function of time. What is the expectation value of the
Solve the time-independent Schrödinger equation for a centered infinite square well with a delta-function barrier in the middle:Treat the even and odd wave functions separately. Don’t bother to
Consider the “step” potential(a) Calculate the reflection coefficient, for the case E < V0, and comment on the answer.(b) Calculate the reflection coefficient for the case E <
Determine the transmission coefficient for a rectangular barrier (same as Equation 2.148, only with V (x) = + V0 > 0 in the region -α < x < a). Treat separately the three cases E < V0, E
Derive Equations 2.170 and 2.171.Plug these back into Equations 2.166 and 2.167. Obtain the transmission coefficient, and confirm Equation 2.172. k k ika [sin (fa) 1 eus (c)] : [ers (ta), sin da)]
Solve the time-independent Schrödinger equation with appropriate boundary conditions for the “centered” infinite square well: V (x) = 0 (for -α < x < + α), V (x) = ∞ (otherwise).
Imagine a bead of mass m that slides frictionlessly around a circular wire ring of circumference L. (This is just like a free particle, except that Ψ (x+L) = Ψ (x).) Find the stationary states
Consider the double delta-function potentialwhere α and a are positive constants.(a) Sketch this potential.(b) How many bound states does it possess? Find the allowed energies, for α = ћ2 /mα and
If two (or more) distinct solutions to the (time-independent) Schrödinger equation have the same energy E, these states are said to be degenerate. For example, the free particle states are doubly
The Dirac delta function can be thought of as the limiting case of a rectangle of area 1, as the height goes to infinity and the width goes to zero. Show that the delta-function well (Equation 2.117)
This is a strictly qualitative problem—no calculations allowed! Consider the “double square well” potential (Figure 2.20). Suppose the depth V0 and the width α are fixed, and large enough so
Find the first three excited state energies (to five significant digits) for the harmonic oscillator, by wagging the dog (Problem 2.55). For the first (and third) excited state you will need to set u
Prove the famous “(your name) uncertainty principle,” relating the uncertainty in position (A = x) to the uncertainty in energyFor stationary states this doesn’t tell you much—why not? (B =
(a) Prove the following commutator identities:(b) Show that(c) Show more generally thatfor any function f(x) that admits a Taylor series expansion.(d) Show that for the simple harmonic oscillatorUse
Show thatIn momentum space, then, the position operator is iћ∂/∂p. More generally,In principle you can do all calculations in momentum space just as well (though not always as easily) as in
Find ϕ (p,t) for the free particle in terms of the function ϕ (k) introduced in Equation 2.101. Show that for the free particle |ϕ (p,t)|2 is independent of time. The time independence of |ϕ
Consider the operatorwhere (as in Example 3.1)ϕ is the azimuthal angle in polar coordinates, and the functions are subject to Equation 3.26. Is hermitian? Find its eigenfunctions and
Find the momentum-space wave function, ϕ (p,t), for a particle in the ground state of the harmonic oscillator. What is the probability (to two significant digits) that a measurement of p on a
(a) Cite a Hamiltonian from Chapter 2 (other than the harmonic oscillator) that has only a discrete spectrum.(b) Cite a Hamiltonian from Chapter 2 (other than the free particle) that has only a
(a) Show that the set of all square-integrable functions is a vector space (refer to Section A.1 for the definition). The main point is to show that the sum of two square-integrable functions is
Find the first four allowed energies (to five significant digits) for the infinite square well, by wagging the dog.
Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Show that if P̂ and Q̂ have a complete set of common eigenfunctions, then [P̂,Q̂] f = 0 for any function
Consider operators Â and B̂ that do not commute with each other but do commute with their commutator: (for instance, x̂ and p̂).(a) Show thatYou can prove this by induction on n,
Let (the derivative operator). Find(a)(b) D = d/dx
Consider a three-dimensional vector space spanned by an orthonormal basis |1>,|2>,|3>. Kets |α> and |β> and are given by(a) Construct <α |and <β |(in terms of the dual basis
Let be an operator with a complete set of orthonormal eigenvectors:(a) Show that can be written in terms of its spectral decomposition:for any vector |α>.(b) Another way to define a
Test the energy-time uncertainty principle for the wave function in Problem 2.5 and the observable x, by calculating σH, σx, and d(x)/dt exactly.Problem 2.5A particle in the infinite square well
Show that the energy-time uncertainty principle reduces to the “your name” uncertainty principle (Problem 3.15), when the observable in question is x. (Problem 3.15)Prove the famous “(your
Apply Equation 3.73 to the following special cases:(a) Q = 1;(b) Q = H;(c) Q = x;(d) Q = p;In each case, comment on the result, with particular reference to Equations 1.27, 1.33, 1.38, and
Solve Equation 3.69 for Ψ(x). (x) and (p) are constants (independent of x). d (-ih-(p)) (p) ia(x - (x)), 4 = dx (3.69)
Legendre polynomials. Use the Gram–Schmidt procedure (Problem A.4) to orthonormalize the functions 1, x,x2, and x3, on the interval -1 ≤ x ≤ 1. You may recognize the results—they are (apart
Derive the transformation from the position-space wave function to the “energy-space” wave function (cn (t)) using the technique of Example 3.9. Assume that the energy spectrum is discrete, and
An anti-hermitian (or skew-hermitian) operator is equal to minus its hermitian conjugate:(a) Show that the expectation value of an anti-hermitian operator is imaginary.(b) Show that the eigenvalues
The most general wave function of a particle in the simple harmonic oscillator potential isShow that the expectation value of position iswhere the real constants C and ϕ are given byThus the
Supportfor constants A and a.(a) Determine A, by normalizing Ψ(x,0).(b) Find (x), (x2), and σx (at time t = 0).(c) Find the momentum space wave function Φ(p,0), and check that it is normalized.(d)

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