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Chapter 2: Problem 5

Quantum potential step A well-known quantum mechanics problem involves a particle of mass \(m\) that encounters a one-dimensional potential step, like this: The particle with initial kinetic energy \(E\) and wavevector \(k_{1}=\sqrt{2 m E} / \hbar\) enters from the left and encounters a sudden jump in potential energy of height \(V\) at position \(x=0\). By solving the Schrödinger equation, one can show that when \(E>V\) the particle may either (a) pass the step, in which case it has a lower kinetic energy of \(E-V\) on the other side and a correspondingly smaller wavevector of \(k_{2}=\sqrt{2 m(E-V)} / \hbar\), or \((b)\) it may be reflected, keeping all of its kinetic energy and an unchanged wavevector but moving in the opposite direction. The probabilities \(T\) and \(R\) for transmission and reflection are given by $$ T=\frac{4 k_{1} k_{2}}{\left(k_{1}+k_{2}\right)^{2}}, \quad R=\left(\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right)^{2} . $$ Suppose we have a particle with mass equal to the electron mass \(m=9.11 \times\) \(10^{-31} \mathrm{~kg}\) and energy \(10 \mathrm{eV}\) encountering a potential step of height \(9 \mathrm{eV}\). Write a Python program to compute and print out the transmission and reflection probabilities using the formulas above.

Short Answer

Expert verified
Write the program as directed to solve for T and R.

Step by step solution

01

Import necessary libraries

Use relevant Python libraries for mathematical calculations. In this case, the 'math' library will be used to handle square root operations.
02

Define constants and variables

Assign values to constants such as the mass of the electron, the energies of the particle and the potential step, and the reduced Planck's constant. Also define the wavevectors using their respective formulas.
03

Calculate wavevectors

Using the formulas provided, calculate the initial wavevector (k1) and the wavevector after the potential step (k2).
04

Calculate transmission probability

T = (4 * k1 * k2) / (k1 + k2)^2
05

Calculate reflection probability

R = ((k1 - k2) / (k1 + k2))^2
06

Print results

Print the calculated values of the transmission and reflection probabilities.

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Key Concepts

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

Quantum Potential Step
The quantum potential step is a fundamental concept in quantum mechanics, showcasing how particles behave when they encounter sudden changes in potential energy. Imagine a particle moving in one dimension and meeting a step-like jump in potential energy. This scenario helps us understand how particles, such as electrons, interact with potential barriers as they move through space.
The exercise considers a particle approaching a step of height \(V\) in potential energy at position \(x=0\). Depending on its initial kinetic energy \(E\), the particle can either pass over the step or be reflected back.
When \(E > V\), the particle has two possibilities:
  • If the particle passes the step, it experiences a decrease in kinetic energy to \(E - V\) and a smaller wavevector. This means it slows down after crossing the step.
  • If the particle is reflected, it retains its initial kinetic energy \(E\) and wavevector but reverses direction.
Understanding these outcomes requires solving the Schrödinger equation, giving us insights into transmission and reflection probabilities, which will be discussed in the following sections.
Schrödinger Equation
The Schrödinger equation is the cornerstone of quantum mechanics, describing how the quantum state of a system evolves over time. For a particle encountering a potential step, we use the time-independent Schrödinger equation:
\[ -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi = E \psi \]
In this equation:
  • \(\hbar\) is the reduced Planck's constant.
  • \(m\) is the mass of the particle.
  • \(V(x)\) is the potential energy at position \(x\).
  • \(E\) is the total energy of the particle.
To solve the Schrödinger equation, we consider the potential step at \(x=0\). The solutions are different for regions \(x < 0\) and \(x \geq 0\):
  • For \(x < 0\), the solution involves an incident wave moving towards the step and a reflected wave moving away from the step.
  • For \(x \geq 0\), the solution describes a transmitted wave moving beyond the step.
By applying boundary conditions at \(x=0\), we match the wave functions and their derivatives on both sides of the step. This leads to expressions for the transmission and reflection coefficients, providing a direct link between the Schrödinger equation and observable probabilities.
Transmission and Reflection Probabilities
The probabilities of transmission \(T\) and reflection \(R\) describe how likely it is for a particle to pass over or be reflected by a potential step. These probabilities are derived from the wavevectors on both sides of the step:
  • Initial wavevector: \(k_1 = \sqrt{\frac{2mE}{\hbar^2}}\).
  • Wavevector after the step: \(k_2 = \sqrt{\frac{2m(E-V)}{\hbar^2}}\).
Using these wavevectors, the transmission and reflection probabilities are calculated as:
\[ T = \frac{4k_1k_2}{(k_1 + k_2)^2} \]
\[ R = \left(\frac{k_1 - k_2}{k_1 + k_2}\right)^2 \]
Here's how we understand these probabilities:
  • Higher transmission probability (\(T\)) means the particle is more likely to pass the step, continuing its journey on the other side.
  • Higher reflection probability (\(R\)) indicates the particle is more likely to bounce back, retaining its initial kinetic energy.
For example, if an electron with energy \(10 eV\) approaches a potential step of \(9 eV\), we can use our Python program to compute \(T\) and \(R\). This helps illustrate how quantum mechanics predicts different outcomes based on the initial conditions of the particle and the potential step it encounters.

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Most popular questions from this chapter

The Madelung constant In condensed matter physics the Madelung constant gives the total electric potential felt by an atom in a solid. It depends on the charges on the other atoms nearby and their locations. Consider for instance solid sodium chloride- table salt. The sodium chloride crystal has atoms arranged on a cubic lattice, but with alternating sodium and chlorine atoms, the sodium ones having a single positive charge \(+e\) and the chlorine ones a single negative charge \(-e\), where \(e\) is the charge on the electron. If we label each position on the lattice by three integer coordinates \((i, j, k)\), then the sodium atoms fall at positions where \(i+j+k\) is even, and the chlorine atoms at positions where \(i+j+k\) is odd. Consider a sodium atom at the origin, \(i=j=k=0\), and let us calculate the Madelung constant. If the spacing of atoms on the lattice is \(a\), then the distance from the origin to the atom at position \((i, j, k)\) is $$ \sqrt{(i a)^{2}+(j a)^{2}+(k a)^{2}}=a \sqrt{i^{2}+j^{2}+k^{2}} $$ and the potential at the origin created by such an atom is $$ V(i, j, k)=\pm \frac{e}{4 \pi e_{0} a \sqrt{i^{2}+j^{2}+k^{2}}}, $$ with \(e_{0}\) being the permittivity of the vacuum and the sign of the expression depending on whether \(i+j+k\) is even or odd. The total potential felt by the sodium atom is then the sum of this quantity over all other atoms. Let us assume a cubic box around the sodium at the origin, with \(L\) atoms in all directions. Then $$ V_{\text {beal }}=\sum_{n, j, i=-t \atop n=0}^{L} V(i, j, k)=\frac{e}{4 \pi e_{0} a} M \text {, } $$ where \(M\) is the Madelung constant, at least approximately-technically the Madelung constant is the value of \(M\) when \(L \rightarrow \infty\), but one can get a good approximation just by using a large value of \(L\). Write a program to calculate and print the Madelung constant for sodium chloride. Use as large a value of \(L\) as you can, while still having your program run in reasonable time-say in a minute or less.

Binomial coefficients The binomial coefficient \(\left(\begin{array}{l}n \\ k\end{array}\right)\) is an integer equal to $$ \left(\begin{array}{l} n \\ k \end{array}\right)=\frac{n !}{k !(n-k) !}=\frac{n \times(n-1) \times(n-2) \times \ldots \times(n-k+1)}{1 \times 2 \times \ldots \times k} $$ when \(k \geq 1\), or \(\left(\begin{array}{l}n \\ 0\end{array}\right)=1\) when \(k=0\). a) Using this form for the binomial coefficient, write a Python user-defined function binomial \((\mathrm{n}, \mathrm{k})\) that calculates the binomial coefficient for given \(n\) and \(k\). Make sure your function returns the answer in the form of an integer (not a float) and gives the correct value of 1 for the case where \(k=0\). b) Using your function write a program to print out the first 20 lines of "Pascal's triangle." The \(n\)th line of Pascal's triangle contains \(n+1\) numbers, which are the coefficients \(\left(\begin{array}{l}n \\\ 0\end{array}\right),\left(\begin{array}{l}n \\ 1\end{array}\right)\), and so on up to \(\left(\begin{array}{l}n \\ n\end{array}\right)\). Thus the first few lines are $$ \begin{aligned} &11 \\ &121 \\ &1331 \\ &14641 \end{aligned} $$ c) The probability that an unbiased coin, tossed \(n\) times, will come up heads \(k\) times is \(\left(\begin{array}{l}n \\ k\end{array}\right) / 2^{n}\). Write a program to calculate (a) the total probability that a coin tossed 100 times comes up heads exactly 60 times, and (b) the probability that it comes up heads 60 or more times.

Another ball dropped from a tower A ball is again dropped from a tower of height \(h\) with initial velocity zero. Write a program that asks the user to enter the height in meters of the tower and then calculates and prints the time the ball takes until it hits the ground, ignoring air resistance. Use your program to calculate the time for a ball dropped from a \(100 \mathrm{~m}\) high tower.

The semi-empirical mass formula In nuclear physics, the semi-empirical mass formula is a formula for calculating the approximate nuclear binding energy \(B\) of an atomic nucleus with atomic number \(Z\) and mass number \(A\) : $$ B=a_{1} A-a_{2} A^{2 / 3}-a_{3} \frac{Z^{2}}{A^{1 / 3}}-a_{4} \frac{(A-2 Z)^{2}}{A}+\frac{a_{5}}{A^{1 / 2}} $$ where, in units of millions of electron volts, the constants are \(a_{1}=15.8, a_{2}=18.3\), \(a_{3}=0.714, a_{4}=23.2\), and $$ a_{5}= \begin{cases}0 & \text { if } A \text { is odd, } \\ 12.0 & \text { if } A \text { and } Z \text { are both even, } \\ -12.0 & \text { if } A \text { is even and } Z \text { is odd. }\end{cases} $$ a) Write a program that takes as its input the values of \(A\) and \(Z\), and prints out the binding energy for the corresponding atom. Use your program to find the binding energy of an atom with \(A=58\) and \(Z=28\). (Hint: The correct answer is around \(500 \mathrm{MeV}\) ) b) Modify your program to print out not the total binding energy B, but the binding energy per nucleon, which is \(B / A\). c) Now modify your program so that it takes as input just a single value of the atomic number \(Z\) and then goes through all values of \(A\) from \(A=Z\) to \(A=3 Z\), to find the one that has the largest binding energy per nucleon. This is the most stable nucleus with the given atomic number. Have your program print out the value of \(A\) for this most stable nucleus and the value of the binding energy per nucleon d) Modify your program again so that, instead of taking \(Z\) as input, it runs through all values of \(Z\) from 1 to 100 and prints out the most stable value of \(A\) for each one. At what value of \(Z\) does the maximum binding energy per nucleon occur? (The true answer, in real life, is \(Z=28\), which is nickel.)

Planetary orbits The orbit in space of one body around another, such as a planet around the Sun, need not be circular. In general it takes the form of an ellipse, with the body sometimes closer in and sometimes further out. If you are given the distance \(\ell_{1}\) of closest approach that a planet makes to the Sun, also called its perihelion, and its linear velocity \(v_{1}\) at perihelion, then any other property of the orbit can be calculated from these two as follows. a) Kepler's second law tells us that the distance \(\ell_{2}\) and velocity \(v_{2}\) of the planet at its most distant point, or aphelion, satisfy \(\ell_{2} v_{2}=\ell_{1} v_{1}\). At the same time the total energy, kinetic plus gravitational, of a planet with velocity \(v\) and distance \(r\) from the Sun is given by $$ E=\frac{1}{2} m v^{2}-G \frac{m M}{r}, $$ where \(m\) is the planet's mass, \(M=1.9891 \times 10^{30} \mathrm{~kg}\) is the mass of the Sun, and \(G=6.6738 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) is Newton's gravitational constant. Given that energy must be conserved, show that \(v_{2}\) is the smaller root of the quadratic equation $$ v_{2}^{2}-\frac{2 G M}{v_{1} \ell_{1}} v_{2}-\left[v_{1}^{2}-\frac{2 G M}{\ell_{1}}\right]=0 . $$ Once we have \(v_{2}\) we can calculate \(\ell_{2}\) using the relation \(\ell_{2}=\ell_{1} v_{1} / v_{2}\). b) Given the values of \(v_{1}, \ell_{1}\), and \(\ell_{2}\), other parameters of the orbit are given by simple formulas can that be derived from Kepler's laws and the fact that the orbit is an ellipse: $$ \begin{aligned} \text { Semi-major axis: } & a=\frac{1}{2}\left(\ell_{1}+\ell_{2}\right), \\ \text { Semi-minor axis: } & b=\sqrt{\ell_{1} \ell_{2}}, \\ \text { Orbital period: } & T=\frac{2 \pi a b}{\ell_{1} v_{1}}, \\ \text { Orbital eccentricity: } & e=\frac{\ell_{2}-\ell_{1}}{\ell_{2}+\ell_{1}} . \end{aligned} $$ Write a program that asks the user to enter the distance to the Sun and velocity at perihelion, then calculates and prints the quantities \(\ell_{2}, v_{2}, T\), and \(e\). c) Test your program by having it calculate the properties of the orbits of the Earth (for which \(\ell_{1}=1.4710 \times 10^{11} \mathrm{~m}\) and \(v_{1}=3.0287 \times 10^{4} \mathrm{~ms}^{-1}\) ) and Halley's comet \(\left(\ell_{1}=8.7830 \times 10^{10} \mathrm{~m}\right.\) and \(\left.v_{1}=5.4529 \times 10^{4} \mathrm{~m} \mathrm{~s}^{-1}\right)\). Among other things, you should find that the orbital period of the Earth is one year and that of Halley's comet is about 76 years.
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