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Ch. 7 Problems

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Basic

Medium

Advanced

Basic

B-1. Commutation Relations of Ladder Operators

Expand the products \(\hat{a}\hat{a}^\dagger\) and \(\hat{a}^\dagger\hat{a}\) of the ladder operators

\(\hat{a} = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat{x} + \frac{i}{m\omega}\hat{p}\right), \qquad \hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat{x} - \frac{i}{m\omega}\hat{p}\right)\)

using only the canonical commutation relation \([\hat{x}, \hat{p}] = i\hbar\), and verify by direct calculation that

\([\hat{a}, \hat{a}^\dagger] = 1, \qquad \hat{H} = \hbar\omega\left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\)

Hint

Carefully expand \(\hat{a}\hat{a}^\dagger = \frac{m\omega}{2\hbar}(\hat{x}^2 + \hat{p}^2/(m\omega)^2 - i[\hat{x},\hat{p}]/(m\omega))\) and substitute \([\hat{x},\hat{p}] = i\hbar\). Confirm both that the difference with \(\hat{a}^\dagger\hat{a}\) equals 1, and that the sum takes the form of the Hamiltonian.

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Medium

M-1. Commutation Relations of the Hamiltonian with Ladder Operators

Using the commutator identity \([\hat{A}\hat{B}, \hat{C}] = \hat{A}[\hat{B}, \hat{C}] + [\hat{A}, \hat{C}]\hat{B}\) and the result from Problem 7.1, \([\hat{a}, \hat{a}^\dagger] = 1\), derive

\([\hat{H}, \hat{a}^\dagger] = +\hbar\omega\,\hat{a}^\dagger, \qquad [\hat{H}, \hat{a}] = -\hbar\omega\,\hat{a}\)

From this, show that if \(\hat{H}|n\rangle = E_n|n\rangle\), then \(\hat{a}^\dagger|n\rangle\) is an eigenstate with energy \(E_n + \hbar\omega\), and \(\hat{a}|n\rangle\) is an eigenstate with energy \(E_n - \hbar\omega\).

Hint

For \(\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + 1/2)\), use \([\hat{a}^\dagger\hat{a}, \hat{a}^\dagger] = \hat{a}^\dagger[\hat{a}, \hat{a}^\dagger] + [\hat{a}^\dagger, \hat{a}^\dagger]\hat{a} = \hat{a}^\dagger\). Then expand \(\hat{H}(\hat{a}^\dagger|n\rangle) = (\hat{a}^\dagger\hat{H} + [\hat{H}, \hat{a}^\dagger])|n\rangle\).

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M-2. Normalized Ladder Operations

Starting from only \(\hat{a}|0\rangle = 0\) and \([\hat{a}, \hat{a}^\dagger] = 1\), derive the following normalized ladder operations:

\(\hat{a}^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle, \qquad \hat{a}|n\rangle = \sqrt{n}\,|n-1\rangle\)

Furthermore, starting from the ground state \(|0\rangle\), verify that the normalization of

\(|n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}}|0\rangle\)

is correct.

Hint

To compute \(\|\hat{a}^\dagger|n\rangle\|^2 = \langle n|\hat{a}\hat{a}^\dagger|n\rangle\), use \(\hat{a}\hat{a}^\dagger = \hat{a}^\dagger\hat{a} + 1 = \hat{N} + 1\). Since \(\hat{N}|n\rangle = n|n\rangle\), we get \(\|\hat{a}^\dagger|n\rangle\|^2 = n + 1\). This gives the normalization constant for \(\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle\). The \(\hat{a}|n\rangle\) case follows similarly.

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M-3. Why the Schrödinger Equation Is Not Lorentz Covariant

(a) Confirm that the free-particle Schrödinger equation \(i\hbar\,\partial_t \psi = -\frac{\hbar^2}{2m}\,\partial_x^2\, \psi\) contains a first-order time derivative on the left-hand side and a second-order spatial derivative on the right-hand side.

(b) Explain that under the Minkowski metric \(\eta_{\mu\nu} = \mathrm{diag}(-1, +1, +1, +1)\), the differential operator that forms a Lorentz scalar is \(\partial^\mu\partial_\mu = -c^{-2}\partial_t^2 + \nabla^2\) (the d'Alembertian), and that the orders of the time and spatial derivatives must match.

(c) As a straightforward candidate for making the Schrödinger equation Lorentz covariant, write down the Klein-Gordon equation obtained by promoting \(E^2 = p^2c^2 + m^2c^4\) to operators:

\(\left(-\frac{1}{c^2}\partial_t^2 + \nabla^2 - \frac{m^2c^2}{\hbar^2}\right)\phi = 0\)

Confirm that this contains a second-order time derivative. State qualitatively why this "second-order" nature directly leads to the appearance of negative-energy solutions.

Hint

(b) Since Lorentz transformations mix time and space, an \(n\)-th order time derivative transforms into an \(m\)-th order spatial derivative. If the orders do not match, the form of the equation changes after the transformation. (c) Substituting the plane wave \(e^{-iEt/\hbar + ipx/\hbar}\) yields \(E^2 = p^2c^2 + m^2c^4\). Since this is a quadratic equation in \(E\), there are two solutions \(E = \pm\sqrt{p^2c^2 + m^2c^4}\). The negative branch corresponds to negative-energy solutions.

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Advanced

A-1. Infinite Harmonic Oscillators and the Zero-Point Energy of a String

In string theory, each vibrational mode of a string (mode number \(n = 1, 2, 3, \ldots\)) behaves as an independent harmonic oscillator, and the angular frequency of each mode is an integer multiple \(\omega_n = n\,\omega_1\) (where \(\omega_1\) is the angular frequency of the fundamental vibration).

(a) Using the results from Section 7.4, explain that the zero-point energy of each mode is \(\hbar\omega_n/2 = n\hbar\omega_1/2\).

(b) Show that the total zero-point energy is formally given by

\(E_{\text{zero}} = \frac{\hbar\omega_1}{2}\sum_{n = 1}^{\infty} n\)

and confirm that this sum diverges.

(c) In zeta function regularization, \(\sum_{n = 1}^{\infty} n\) is replaced by \(\zeta(-1) = -\frac{1}{12}\). Evaluate \(E_{\text{zero}}\) under this prescription and show that the result is negative. Note that this is one of the origins of the critical dimension \(D = 26\) for the bosonic string discussed in Ch. 14.

Hint

(a) Simply substitute \(\omega \to \omega_n = n\omega_1\) into the result \(E_0 = \hbar\omega/2\) derived in Section 7.4. (b) Write out the sum explicitly. (c) Zeta function regularization is a prescription that "assigns a finite value to a divergent sum via analytic continuation." Mathematical rigor is deferred to the Appendix (or the renormalization discussion in Quantum Field Theory Quantum Field Theory Ch. 6), but here one simply accepts the prescription and substitutes. \(E_{\text{zero}} = -\hbar\omega_1/24\).

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A-2. Divergent Sum of Zero-Point Energy and the Critical Dimension D = 26

Each transverse oscillation mode of the bosonic string (\(D - 2\) transverse degrees of freedom, mode number \(n = 1, 2, 3, \ldots\)) behaves as an independent harmonic oscillator. The total zero-point energy can be formally written as

\[E_{\text{zero}} = (D - 2) \cdot \frac{\hbar\omega_1}{2}\sum_{n=1}^{\infty} n\]

(where \(\omega_1\) is the angular frequency of the fundamental mode).

(a) After confirming that this sum diverges, apply zeta function regularization \(\sum_{n=1}^{\infty} n \to \zeta(-1) = -\frac{1}{12}\) to express \(E_{\text{zero}}\) as a finite value.

(b) In light-cone quantization, show that the mass-shell condition (Virasoro constraint \(L_0 = 1\)) yields the ground state mass as

\[\alpha' m^2 = \frac{D - 2}{24} - 1\]

(where the regularized zero-point energy appears as \(a = (D-2)/24\)).

(c) After confirming that the ground state is a tachyon (\(m^2 < 0\)), accept that Lorentz invariance requires \(m^2 = 0\) (i.e., \(a = 1\)) for the first excited state to be a massless vector particle, and derive \(D = 26\).

Hint

(b) In light-cone quantization, there are \(D - 2\) physical transverse directions. Each mode \(n\) in each direction carries zero-point energy \(\hbar\omega_n/2\). The regularized ordering constant is \(a = -\frac{D-2}{2}\zeta(-1) = -\frac{D-2}{2}\cdot(-\frac{1}{12}) = (D-2)/24\). (c) Set \(a = 1\) and solve \((D-2)/24 = 1\).