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Chapter 1: Problem 8

Suppose you add a constant \(V_{0}\) to the potential energy (by "constant" I mean independent of \(x\) as well as \(t\) ). In classical mechanics this doesn't change anything, but what about quantum mechanics? Show that the wave function picks up a time-dependent phase factor: \(\exp \left(-i V_{0} t / \hbar\right) .\) What effect does this have on the expectation value of a dynamical variable?

Short Answer

Expert verified
The wave function acquires a time-dependent phase factor, but this phase has no effect on observable quantities such as expectation values.

Step by step solution

01

Hamiltonian with Constant Potential

In quantum mechanics, the Hamiltonian operator describes the total energy of a system. If we add a constant term, \(V_0\), to the potential energy, the Hamiltonian becomes \(\hat{H}' = \hat{H} + V_0\), where \(\hat{H}\) is the original Hamiltonian.
02

Time Evolution of the Wave Function

The time evolution of a quantum state \(\psi(x,t)\) is given by the Schr枚dinger equation:\[i \frac{d\psi(x,t)}{dt} = -\frac{i}{\hbar} \hat{H}' \psi(x,t).\]Substituting \(\hat{H}' = \hat{H} + V_0\), we get:\[\frac{d\psi(x,t)}{dt} = -\frac{i}{\hbar} (\hat{H} + V_0) \psi(x,t) .\]
03

Solving the Modified Schr枚dinger Equation

The solution to the above differential equation is:\[\psi'(x,t) = e^{-i V_0 t / \hbar} \psi(x,t),\]where \(\psi(x,t)\) is the solution to the Schr枚dinger equation without the constant \(V_0\). The wave function picks up an additional time-dependent phase factor \(e^{-i V_0 t / \hbar}\).
04

Effect on Expectation Value

The expectation value of an operator \(\hat{A}\) is given by:\[\langle \hat{A} \rangle = \int \psi'^*(x,t) \hat{A} \psi'(x,t) \;dx.\]Substituting \(\psi'(x,t) = e^{-i V_0 t / \hbar} \psi(x,t)\), we get:\[\langle \hat{A} \rangle = \int (e^{i V_0 t / \hbar} \psi^*(x,t)) \hat{A} (e^{-i V_0 t / \hbar} \psi(x,t)) \;dx.\]The exponential phase factors cancel each other, leaving:\[\langle \hat{A} \rangle = \int \psi^*(x,t) \hat{A} \psi(x,t) \;dx,\]meaning the expectation value remains unchanged.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Phase Factor
In quantum mechanics, the concept of a phase factor plays a crucial role in understanding how quantum states evolve over time. When a constant potential energy, denoted as \(V_0\), is added to a quantum mechanical system, the wave function acquires a phase factor of the form \(\exp \left(-i \frac{V_0 t}{\hbar} \right)\). This is a mathematical expression representing a rotation in the complex plane.

  • The phase factor is complex, with the 'i' representing the imaginary unit.
  • \(t\) is the time variable, and \(\hbar\) is the reduced Planck's constant, a fundamental quantity in quantum physics.
  • This factor does not alter the probability density, which depends only on the magnitude of the wave function.
In essence, while the phase factor can affect interference patterns in quantum mechanics, it does not affect measurable probabilities or outcomes.
Expectation Value
The expectation value is a core concept in quantum mechanics used to predict the average outcome of a large number of measurements of a particular observable. It is important to understand how adding a phase factor to a wave function impacts the expectation value of an observable represented by an operator \(\hat{A}\).

When a phase factor \(\exp \left(-i \frac{V_0 t}{\hbar} \right)\) is introduced:
  • The expectation value formula involves integrating the wave function and its complex conjugate over all space.
  • Due to the properties of exponential functions, the phase factors cancel each other out when calculating the expectation value.
  • This leaves the expression for the expectation value unchanged.

    This result demonstrates that though the wave function itself changes by acquiring the phase factor, the statistical predictions for observables remain unaffected.
Hamiltonian Operator
The Hamiltonian operator, \(\hat{H}\), is essential in quantum mechanics. It represents the total energy of the system, including both kinetic and potential energies. When a constant \(V_0\) is added, the Hamiltonian changes to \(\hat{H}' = \hat{H} + V_0\). Key points about this operator include:
  • The original Hamiltonian, \(\hat{H}\), dictates the natural time evolution of the system.
  • Adding a constant potential energy, \(V_0\), modifies the Hamiltonian but only adds a phase factor to the wave function.
  • This adjustment retains the form of the Schr枚dinger equation but modifies its solution to include the time-dependent phase factor \(\exp \left(-i \frac{V_0 t}{\hbar} \right)\).
Overall, the Hamiltonian operator ensures that even when modified by a constant, the physical essence of the observable quantities remains unaltered.

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Most popular questions from this chapter

Suppose you wanted to describe an unstable particle, that spontaneously disintegrates with a "lifetime" T. In that case the total probability of finding the particle somewhere should not be constant, but should decrease at (say) an exponential rate: $$P(t) \equiv \int_{-\infty}^{+\infty}|\Psi(x, t)|^{2} d x=e^{-t / \tau}.$$ A crude way of achieving this result is as follows. In Equation 1.24 we tacitly assumed that \(V(\) the potential energy) is real. That is certainly reasonable, but it leads to the "conservation of probability" enshrined in Equation \(1.27 .\) What if we assign to \(V\) an imaginary part: $$V=V_{0}-i \Gamma,$$ where \(V_{0}\) is the true potential energy and \(\Gamma\) is a positive real constant? (a) Show that (in place of Equation 1.27 ) we now get $$\frac{d P}{d t}=-\frac{2 \Gamma}{\hbar} P$$. (b) Solve for \(P(t)\), and find the lifetime of the particle in terms of \(\Gamma\).

Consider the wave function $$\Psi(x, t)=A e^{-\lambda|x|} e^{-i \omega t},$$ where \(A, \lambda,\) and \(\omega\) are positive real constants. (We'll see in Chapter 2 for what potential \((V)\) this wave function satisfies the Schr枚dinger equation.) (a) Normalize \(\Psi\). (b) Determine the expectation values of \(x\) and \(x^{2}\). (c) Find the standard deviation of \(x\). Sketch the graph of \(|\Psi|^{2},\) as a function of \(x,\) and mark the points \((\langle x\rangle+\sigma)\) and \((\langle x\rangle-\sigma),\) to illustrate the sense in which \(\sigma\) represents the "spread" in \(x\). What is the probability that the particle would be found outside this range?

A particle of mass \(m\) has the wave function $$\Psi(x, t)=A e^{-a\left[\left(m x^{2} / \hbar\right)+i t\right]},$$ where \(A\) and \(a\) are positive real constants. (a) Find \(A\). (b) For what potential energy function, \(V(x)\), is this a solution to the Schr枚dinger equation? (c) Calculate the expectation values of \(x, x^{2}, p,\) and \(p^{2}\). (d) Find \(\sigma_{x}\) and \(\sigma_{p} .\) Is their product consistent with the uncertainty principle?

[This problem generalizes Example 1.2.] Imagine a particle of mass \(m\) and energy \(E\) in a potential well \(V(x),\) sliding frictionlessly back and forth between the classical turning points ( \(a\) and \(b\) in Figure 1.10 ). Classically, the probability of finding the particle in the range \(d x\) (if, for example, you took a snapshot at a random time \(t\) ) is equal to the fraction of the time \(T\) it takes to get from \(a\) to \(b\) that it spends in the interval \(d x\): $$\rho(x) d x=\frac{d t}{T}=\frac{(d t / d x) d x}{T}=\frac{1}{v(x) T} d x,$$ where \(v(x)\) is the speed, and $$T=\int_{0}^{T} d t=\int_{a}^{b} \frac{1}{v(x)} d x.$$ Thus $$\rho(x)=\frac{1}{v(x) T}.$$ This is perhaps the closest classical analog to \(|\Psi|^{2}\). (a) Use conservation of energy to express \(v(x)\) in terms of \(E\) and \(V(x)\). (b) As an example, find \(\rho(x)\) for the simple harmonic oscillator, \(V(x)=k x^{2} / 2 .\) Plot \(\rho(x),\) and check that it is correctly normalized. (c) For the classical harmonic oscillator in part (b), find \(\langle x\rangle,\left\langle x^{2}\right\rangle,\) and \(\sigma_{x}\).

Consider the first 25 digits in the decimal expansion of \(\pi(3,1,4,1,5,9, \dots)\). (a) If you selected one number at random, from this set, what are the probabilities of getting each of the 10 digits? (b) What is the most probable digit? What is the median digit? What is the average value? (c) Find the standard deviation for this distribution.
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