Question

Show that for scattering problems \((E>0)\) (a) the integral form of the Schrödinger equation in one dimension is $$ \Psi(x)=e^{i k x}-\frac{i \mu}{\hbar^{2} k} \int_{-\infty}^{\infty} e^{i k|x-y|} V(y) \Psi(y) d y . $$ (b) Divide \((-\infty,+\infty)\) into three regions \(R_{1}=(-\infty,-a), R_{2}=(-a,+a)\) and \(R_{3}=(a, \infty)\). Let \(\psi_{i}(x)\) be \(\psi(x)\) in region \(R_{i} .\) Assume that the potential \(V(x)\) vanishes in \(R_{1}\) and \(R_{3}\). Show that $$ \psi_{1}(x)=e^{i k x}-\frac{i \mu}{\hbar^{2} k} e^{-i k x} \int_{-a}^{a} e^{i k y} V(y) \psi_{2}(y) d y $$ This shows that determining the wave function in regions where there is no potential requires the wave function in the region where the potential acts. (c) Let $$ V(x)=\left\\{\begin{array}{ll} V_{0} & \text { if }|x|a \end{array}\right. $$ and find \(\psi_{2}(x)\) by the method of successive approximations. Show that the \(n\) th term is less than \(\left(2 \mu V_{0} a / \hbar^{2} k\right)^{n-1}\) (so the Neumann series will converge) if \(\left(2 V_{0} a / \hbar v\right)<1\), where \(v\) is the velocity and \(\mu v=\) \(\hbar k\) is the momentum of the wave. Therefore, for large velocities, the Neumann series expansion is valid.

Short answer

The problem is broken into three tasks, first requiring proving the given equation is the integral form of the Schrödinger equation. The second task, with appropriately defined regions, yields to an equation for \(\psi_1(x)\). Finally, under the given conditions, we can find \(\psi_2(x)\) using successive approximations. This iteration results in showing the convergence of the Neumann series for large velocities.

Step-by-step solution

Prove the Integral form of the Schrödinger Equation

To show the given equation is the integral form of the Schrödinger equation in one dimension, it’s necessary to use the concept of Green’s function, where we consider the integral form of the wave equation in conjunction with the Green’s function.

Solve for \(\psi_1(x)\)

The potential \(V(x)\) is zero in \(R_1\) and \(R_3\), which would make the integral in the formula for \(\Psi(x)\) disappear. This simplification yields the equation in step 2.

Evaluate \(\psi_2(x)\) using Successive Approximations

Use the method of successive approximations to compute \(\psi_2(x)\), where the integral form of the Schrödinger equation is applied repeatedly. This method essentially involves taking an initial approximation for the wave function and then continually substituting this into the integral on the right hand side of the equation to get successively better approximations.

Show convergence of the series

Using the properties and assumptions of \(V(x)\) and the wave function derived, the nth term of the series can be shown to be less than the given function, which confirms that the Neumann series will converge under the given conditions.

Key concepts

Green's function

A Green's function is a powerful mathematical form used to solve inhomogeneous differential equations subject to certain boundary conditions. In the context of one-dimensional quantum mechanics, it helps to uncover how a particle behaves when interacting with an external potential. By expressing the Schrödinger equation in its integral form, we can explore the resultant wave function that governs the particle's state. The integral that includes the Green's function essentially sums over all possible ways the particle can 'feel' the potential, encapsulating the totality of paths into the evolution of the wave function.

When analyzing scattering problems, the Green's function models how an initial wave is modified by the potential, producing a scattered wave that includes both incoming and outgoing components. This function behaves much like a translator, converting the potential landscape into the language of waves, mapping the forces the particle experiences directly to the final shape of its wave function.

Successive approximations

The technique of successive approximations, also known as iterative methods, is a cornerstone in solving complex physics problems when exact solutions are challenging to find. For quantum mechanical problems, where the wave function can be significantly altered due to a potential, this approach allows us to get closer and closer to the correct description step by step. Starting with a baseline guess, successive approximations refine this guess by repeatedly using it within the integral form of the Schrödinger equation.

Each cycle of approximation leverages the most recent guess to calculate the next, ideally honing in on the true wave function with each iteration. From the standpoint of scattering problems, this process incrementally adapts the free particle wave function to account for the dynamic changes induced by the potential, gradually building up the correct scattered pattern.

Neumann series convergence

Convergence is a critical condition for the success of any series approximation method, including Neumann series used in quantum mechanics. Neumann series are analogous to a mathematical rescue rope, letting us climb towards the true solution through a succession of interconnected segments. For convergence to occur, each added 'segment' or term must become successively smaller, ensuring that the series doesn't diverge to infinity.

In the context of scattering problems, if the nth term of a Neumann series is demonstrably smaller than a certain factor raised to the power of n-1, we can be confident that the series will converge to a finite value, which corresponds to the true wave function. This condition is particularly met when we deal with particles that are very fast, meaning that their stored energy, compared to the potential they are interacting with, ensures that each iterative correction to the wave function is smaller than the last, and a true solution can be reached.

Integral form of the wave equation

The integral form of the wave equation is a reformulation of the Schrödinger equation which encompasses the entire influence of the potential on the wave function. Rather than a local, differential description, this form offers a global view of how a particle’s wave function emerges from its interaction with the environment.

In solving scattering problems, the integral form becomes especially insightful. It allows us to see how particles propagate from all points in space, subject to the potential, revealing the resulting scattered wave. Integral equations of this nature are often solved with the help of Green’s functions and iterative methods, enabling physicists to write down solutions that incorporate the full complexity of quantum interactions in a manageable mathematical format.

One-dimensional quantum mechanics

One-dimensional quantum mechanics distills the richer, more complex three-dimensional problems down to a simpler context where particles are constrained to move along a single line. This simplification allows us to tackle fundamental concepts without the mathematical overhead of additional spatial dimensions. In one-dimensional systems, wave functions and probabilities become functions of a single position variable, x, allowing easier visualization and understanding of quantum phenomena such as tunneling, quantization of energy, and the probabilistic nature of measurement outcomes.

Scattering problems in one-dimension prominently feature in the study of quantum mechanics because they set the stage for understanding more involved quantum systems, and they often yield to simpler analytical solutions. They provide a gateway to grasping the translational behavior of a quantum particle when it encounters a potential difference, such as an electron hitting a barrier or a quantum tunneling scenario, showcasing the foundational principles of wave-particle duality.