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Magazine Chapter 2: Problem 44
Solve the time-independent Schr枚dinger equation for a centered infinite square
well with a delta-function barrier in the middle:
$$
V(x)=\left\\{\begin{array}{ll}
\alpha \delta(x), & \text { for }-a
Short Answer
Expert verified
Odd solutions energy remains unchanged; even solutions are shifted. For \(\alpha=0\), it's the standard well; \(\alpha=\infty\), even states disappear.
Step by step solution
01
Identify the Region and Boundary Conditions
The potential is zero for \(-a < x < a\) and infinite elsewhere. The boundary conditions require that the wave function vanish at \(x = \pm a\), ensuring an infinite potential outside is a wall.
02
Set Up the Schr枚dinger Equation Inside the Well
Inside the well, the time-independent Schr枚dinger equation is given by: \(-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi\), where \(0 < x < a\). This simplifies to \(\frac{d^2 \psi}{dx^2} = -k^2 \psi\), with \(k^2 = \frac{2mE}{\hbar^2}\). This equation has solutions \(\psi(x) = A \cos(kx) + B \sin(kx)\).
03
Apply the Even Condition
For even wave functions, \(\psi(-x) = \psi(x)\), thus \(\psi(x) = A \cos(kx)\). Applying boundary conditions at \(x = a\), \(\psi(a) = A \cos(ka) = 0\). This implies \(\cos(ka) = 0\), leading to \(ka = (2n + 1)\frac{\pi}{2}, n \in \mathbb{Z}\).
04
Apply the Odd Condition
Odd wave functions satisfy \(\psi(-x) = -\psi(x)\), implying \(\psi(x) = B \sin(kx)\). Again bounding at \(x = a\), \(\psi(a) = B \sin(ka) = 0\), leading to \(ka = n\pi, n \in \mathbb{Z}\). These solutions are unaffected by the delta barrier.
05
Delta Function Discontinuity Condition
At \(x=0\), \(\psi'\) must have a discontinuity. Integrating the Schr枚dinger equation around \(x=0\), \(\psi'(0^+) - \psi'(0^-) = \frac{2m\alpha}{\hbar^2} \psi(0)\). For even states, \(\psi(0) = A eq 0\); for odd states, \(\psi(0) = 0\) results no effect from the delta.
06
Solve for Even State Energies
The energy equation for even states becomes \(k \tan(ka) = \frac{m\alpha}{\hbar^2}\). Solve graphically or numerically, resulting in modified energies due to the delta function.
07
Compare with Delta Function Absent
Without the delta barrier, energies satisfy \(ka = n\pi\), for both even (\(n>0\)) and odd solutions, leading to unaffected odd solutions and shifted even solutions.
08
Analyze Limiting Cases
For \(\alpha \to 0\), the system reverts to a normal infinite square well, while 路\(\alpha \to \infty\) diverges, forcing even solutions to nullify around the barrier.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Infinite Square Well
The infinite square well is a fundamental quantum mechanics problem. Imagine a particle trapped in a one-dimensional box, but unlike a normal box, it can't escape past the walls. The walls are impenetrable and represented by infinite potential energy. Inside the well, the potential energy is zero, allowing a wave function to describe the particle鈥檚 position.
For the quantum particle, its allowed positions are determined by boundary conditions鈥攕pecifically that the wave function must vanish at the walls of the well. These boundary conditions arise because particles can't exist in regions of infinite potential. These constraints lead to quantized solutions. The particle inside the well can only have energies and shapes that fit perfectly within this box.
The wave functions are sinusoidal (sine or cosine waves) because they satisfy the Schr枚dinger equation under these boundary conditions. The continuity and smoothness of the wave function lead to quantization of energy levels.
For the quantum particle, its allowed positions are determined by boundary conditions鈥攕pecifically that the wave function must vanish at the walls of the well. These boundary conditions arise because particles can't exist in regions of infinite potential. These constraints lead to quantized solutions. The particle inside the well can only have energies and shapes that fit perfectly within this box.
The wave functions are sinusoidal (sine or cosine waves) because they satisfy the Schr枚dinger equation under these boundary conditions. The continuity and smoothness of the wave function lead to quantization of energy levels.
Delta Function Barrier
A delta function barrier is a unique quantum potential representing an "infinitely thin" and "infinitely high" obstacle at a specific point, affecting the particle鈥檚 wave function at that point. In practical terms, it acts as a sharp spike or bump in potential energy that disrupts the continuity of the wave function鈥檚 derivative.
In the problem of an infinite square well with a delta function barrier right in the middle, this disruption imposes an additional condition. The wave function remains continuous, but its derivative experiences a discontinuity. This is mathematically described by the "jump" condition in the derivative of the wave function at the location of the delta function.
The impact on the wave function varies based on its symmetry (even or odd). In particular, for even wave functions, this barrier modifies the energy levels because the wave function's value is non-zero at the barrier, causing an impact. Odd wave functions, however, do not get affected by the barrier since they inherently have a nodal zero at the center鈥攖he same as the position of the barrier.
In the problem of an infinite square well with a delta function barrier right in the middle, this disruption imposes an additional condition. The wave function remains continuous, but its derivative experiences a discontinuity. This is mathematically described by the "jump" condition in the derivative of the wave function at the location of the delta function.
The impact on the wave function varies based on its symmetry (even or odd). In particular, for even wave functions, this barrier modifies the energy levels because the wave function's value is non-zero at the barrier, causing an impact. Odd wave functions, however, do not get affected by the barrier since they inherently have a nodal zero at the center鈥攖he same as the position of the barrier.
Even and Odd Wave Functions
In quantum mechanics, wave functions can be classified based on symmetry as even or odd. An even wave function remains unchanged when flipped about an axis, while an odd wave function acquires a sign change.
For a particle in an infinite square well bisected by a delta barrier, even wave functions have the form of cosine waves. They are symmetric about the origin. As a result, any barriers placed symmetrically in the well, such as at the center, directly affect them. Since the barrier is centered, these wave functions encounter the delta at their maximum, directly altering their energy states.
In contrast, odd wave functions follow the sine wave pattern and are antisymmetric. They naturally have a node at the center of the well. This point of zero amplitude results in no change or interaction with the delta barrier, as the wave function value there is zero, making these odd states immune to the barrier鈥檚 effects.
For a particle in an infinite square well bisected by a delta barrier, even wave functions have the form of cosine waves. They are symmetric about the origin. As a result, any barriers placed symmetrically in the well, such as at the center, directly affect them. Since the barrier is centered, these wave functions encounter the delta at their maximum, directly altering their energy states.
In contrast, odd wave functions follow the sine wave pattern and are antisymmetric. They naturally have a node at the center of the well. This point of zero amplitude results in no change or interaction with the delta barrier, as the wave function value there is zero, making these odd states immune to the barrier鈥檚 effects.
Energy Eigenvalues
In the infinite square well problem, energy eigenvalues are the specific energy levels a particle can occupy. They are solutions to the Schr枚dinger equation under the given boundary conditions.
With no delta function, energy eigenvalues for even and odd solutions can be expressed by the quantum number, known as "n," and are determined by the principle that the wave function must fit an integer number of half-wavelengths within the well.
The presence of a delta function shifts these energy eigenvalues for even solutions. Because the wave function at the center is non-zero for even states, they experience an effective potential due to the delta barrier. The new quantized energies can be calculated with an additional mathematical condition involving the tangent function and other constants like the mass of the particle, \(m\), and Planck's constant, \(\hbar\).
Odd states, however, experience no change in energy because the delta function coincides with their node at the center, where they inherently possess zero probability amplitude. Therefore, the odd solutions retain their energy as if the delta barrier wasn't present.
With no delta function, energy eigenvalues for even and odd solutions can be expressed by the quantum number, known as "n," and are determined by the principle that the wave function must fit an integer number of half-wavelengths within the well.
The presence of a delta function shifts these energy eigenvalues for even solutions. Because the wave function at the center is non-zero for even states, they experience an effective potential due to the delta barrier. The new quantized energies can be calculated with an additional mathematical condition involving the tangent function and other constants like the mass of the particle, \(m\), and Planck's constant, \(\hbar\).
Odd states, however, experience no change in energy because the delta function coincides with their node at the center, where they inherently possess zero probability amplitude. Therefore, the odd solutions retain their energy as if the delta barrier wasn't present.
