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Chapter 2: Problem 46

Consider the potential $$ V(x)=\left\\{\begin{array}{ll} \infty, & \text { if } x<0 \\ \alpha \delta(x-a), & \text { if } x \geq 0 \end{array}\right. $$ where \(a\) and \(\alpha\) are positive real constants with the appropriate units (see Figure 2.18). A particle starts out in the "well" \((0

Short Answer

Expert verified
Solve Schr枚dinger equation in the well. Energy involves complex terms due to tunneling. Lifetime of state relates to imaginary part of energy.

Step by step solution

01

Define the Schr枚dinger Equation

The time-independent Schr枚dinger equation for a particle in a potential is given by \[\frac{-\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi(x) = E \psi(x).\] For the region \(0 < x < a\), the potential \(V(x) = 0\). For \(x > a\), the potential \(V(x) = \alpha \delta(x-a)\). Outside the well, the wave function vanishes, so we only need to solve within the well up to just past \(x = a\).
02

Solve Schr枚dinger's Equation in the Region (0, a)

For \(0 < x < a\), with \(V(x) = 0\), the Schr枚dinger equation simplifies to \[\frac{-\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi(x).\]This is a basic differential equation with solutions of the form \(\psi(x) = A \sin(kx) + B \cos(kx)\) where \(k = \sqrt{\frac{2mE}{\hbar^2}}\). Due to potential boundary conditions later, solutions simplify to appropriate sine or cosine terms.
03

Apply Boundary Conditions

The boundary conditions are \(\psi(0) = 0\) because the potential is infinite at \(x < 0\). Therefore, \(B = 0\) and \(\psi(x) = A \sin(kx)\). At \(x=a\), apply the continuity and discontinuity conditions of the wave function and its derivative at the delta potential. For the delta function, \(\psi(a^+) - \psi(a^-) = 0\) shows continuity, while the change in slope is governed by \[\frac{d\psi}{dx}\bigg|_{a^+} - \frac{d\psi}{dx}\bigg|_{a^-} = \frac{2m\alpha}{\hbar^2} \psi(a).\]
04

Determine the Implicit Energy Equation

Using continuity \( \psi(a^-) = \psi(a^+) \) and applying the discontinuity in the derivative, combine these to form an equation. Substituting from the wave function \( \psi(x) = A\sin(ka) \), the energy condition becomes \[k\cot(ka) = -\frac{m\alpha}{\hbar^2 k}.\] This implicit equation determines the energy eigenvalues where \(k = \sqrt{\frac{2mE}{\hbar^2}}\).
05

Explain the Complex Energy

The presence of a complex energy suggests that the system is open, with \(E = E_0 + i \Gamma\), where the imaginary \(\Gamma\) component accounts for decay of probability due to tunneling through the barrier. The complex energy arises due to the non-conserved probability in an open system. This matches quantum mechanics principles whereby probabilities can 'leak' or decrease over time.
06

Calculate the Time Constant

Using \(\Gamma\), the duration for probability reduction per tunneling is determined via \(\gamma\) as the imaginary part of the energy. The lifetime \(\tau\) to drop to \(1/e\) of its initial probability is \[\tau = \frac{\hbar}{2 \Gamma}.\]This represents the characteristic time the particle remains before significant tunneling occurs.

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Key Concepts

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

Delta Potential
The delta potential is a fascinating concept in quantum mechanics often used to model simple barriers and wells. It is represented mathematically by the delta function, \[ V(x) = \alpha \delta(x-a) \] where \( \alpha \) is a constant and \( \delta(x-a) \) indicates a localized potential spike at position \( x = a \). This potential can simulate barriers that a particle encounters during its movement. In regions where the potential is described by a delta function, the wave function of a particle can experience a sharp change in its slope.
  • The wave function remains continuous, even at the location of the delta potential. This means that the height of the wave function at \( x = a^- \) is the same as at \( x = a^+ \).
  • However, the derivative of the wave function, which relates to the slope, experiences a discontinuity. This discontinuity is a direct consequence of the delta potential.
The potential's influence becomes apparent when exploring quantum tunneling, as these kinds of potentials can evolve systems with interesting tunneling behaviors, unlike classical particles, which would be reflected off barriers.
Complex Energy
Complex energy arises in quantum mechanics when dealing with systems where the probability of finding a particle in a certain region is non-conserved. This is especially evident in tunneling scenarios where quantum particles exhibit probabilities of penetrating potential barriers.In our scenario, the energy \( E \) is expressed as a complex quantity: \[ E = E_0 + i \Gamma \] Here, \( E_0 \) is the real part, representing the measurable energy of the system, and \( i \Gamma \) is the imaginary part, accounting for the decay of the probability amplitude.
  • The degree to which the energy is complex relates to how likely a particle is to "leak" or tunnel through potential barriers.
  • The imaginary component, \( \Gamma \), quantifies the decay rate of the wave function's amplitude over time, indicating how swiftly the particle exits its confined space.
When you solve the Schr枚dinger equation and encounter a complex energy, it's a reflection of an open system where probabilities are lost over time due to tunneling.
Schr枚dinger Equation
The Schr枚dinger equation lies at the heart of quantum mechanics, describing how the quantum state of a physical system changes in time. The time-independent version is crucial for understanding stationary states and is given by: \[\frac{-\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi(x) = E \psi(x). \] In this equation:
  • \(\psi(x)\) is the wave function that holds probability information about the location of a particle.
  • \(V(x)\) represents the potential energy of the particle as a function of position \(x\).
  • \(E\) is the energy eigenvalue associated with the quantum state described by \(\psi(x)\).
The Schr枚dinger equation in a system experiencing delta potential supports the calculation of wave function continuity and slope discontinuity at the potential spike. When attempting to solve this equation:
  • You must apply correct boundary conditions, such as noting \(\psi(x)\) should vanish where potentials are infinitely high.
  • The equation must be solved separately for each region around the potential, ensuring continuity and appropriate slope changes at boundaries.
Understanding how to apply the Schr枚dinger equation in varied potential landscapes is key to mastering quantum mechanics principles like wave function behavior and tunneling.

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Most popular questions from this chapter

In this problem we explore some of the more useful theorems (stated without proof) involving Hermite polynomials. (a) The Rodrigues formula states that $$ H_{n}(\xi)=(-1)^{n} e^{\xi^{2}}\left(\frac{d}{d \xi}\right)^{n} e^{-\xi^{2}} $$ Use it to derive \(H_{3}\) and \(H_{4}\).

A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: $$ \Psi(x, 0)=A\left[\psi_{1}(x)+\psi_{2}(x)\right] . $$ (a) Normalize \(\Psi(x, 0)\). (That is, find \(A\). This is very easy if you exploit the orthonormality of \(\psi_{1}\) and \(\psi_{2}\). Recall that, having normalized \(\Psi\) at \(t=0\), you can rest assured that it stays normalized - if you doubt this, check it explicitly after doing part b.) (b) Find \(\Psi(x, t)\) and \(|\Psi(x, t)|^{2}\). (Express the latter in terms of sinusoidal functions of time, eliminating the exponentials with the help of Euler's formula: \(e^{i \theta}=\) \(\cos \theta+i \sin \theta .)\) Let \(\omega \equiv \pi^{2} \hbar / 2 m a^{2}\) (c) Compute \((x) .\) Notice that it oscillates in time. What is the frequency of the oscillation? What is the amplitude of the oscillation? (If your amplitude is greater than \(a / 2\), go directly to jail.) (d) Compute \(\langle p\rangle\). (As Peter Lorre would say, "Do it ze kveek vay, Johnny!") (e) Find the expectation value of \(H\). How does it compare with \(E_{1}\) and \(E_{2}\) ? (f) A classical particle in this well would bounce back and forth between the walls. If its energy is equal to the expectation value you found in (e), what is the frequency of the classical motion? How does it compare with the quantum frequency you found in (c)?

Evaluate the following integrals: (a) \(\int_{-3}^{+1}\left(x^{3}-3 x^{2}+2 x-1\right) \delta(x+2) d x\) (b) \(\int_{0}^{\infty}[\cos (3 x)+2] \delta(x-\pi) d x\) (c) \(\int_{-1}^{+1} \exp (|x|+3) \delta(x-2) d x\).

Consider the double delta-function potential $$ V(x)=-\alpha[\delta(x+a)+\delta(x-a)] \text { . } $$ where \(\alpha\) and \(a\) are positive constants. (a) Sketch this potential. (b) How many bound states does it possess? Find the allowed energies, for \(\alpha=\) \(\hbar^{2} / m a\) and for \(\alpha=\hbar^{2} / 4 m a\), and sketch the wave 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? Hint: Look in a math table under "Normal Distribution" or "Error Function".
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