The objective of this article is to graphically illustrate to the students the Gaussian wave packet and its momentum representation using spreadsheets. Gaussian wave packet with different standard deviations is obtained using interactive spreadsheet. Using Fourier transform the momentum representation is obtained. Using Gaussian wave function and its momentum representation the uncertainties in position and wave number are obtained the uncertainty relation is verified. This will make the concept, which so far is left to the abstract imagination come alive for the student, and initiate a deeper understanding of quantum physics problems.
The objective of this article is to graphically illustrate to the students the Gaussian wave packet and its momentum representation using spreadsheets. Gaussian wave packet with different standard deviations is obtained using interactive spreadsheet. Using Fourier transform the momentum representation is obtained. Using Gaussian wave function and its momentum representation the uncertainties in position and wave number are obtained the uncertainty relation is verified. This will make the concept, which so far is left to the abstract imagination come alive for the student, and initiate a deeper understanding of quantum physics problems.
Abstract Concept of wave packet is one of the most relevant topic of quantum mechanics but teaching it in an abstract way does not enhance the learning of students. On the other hand visualizations make such topics very easy in classroom learning. To capture the attention of students in classroom teaching, a unique way to simulate the wave packet associated with a localized particle is presented in this paper. This work describes an easy method to generate data for tracing a wave packet by organising an excel spreadsheet. This method is simple and aids the understanding wave packet modulation with variation in range of wave vectors of constituents matter waves. The obtained uncertainty product in position and momentum of the particle can be further utilized to verify Heisenberg’s uncertainty principle.
Using the time-dependent annihilation and creation operators, the invariant operators, for a free mass and an oscillator, we find the coherent-squeezed state representation of a travelling general Gaussian wave packet with initial expectation values, x0 and p0, of the position and momentum and variances, Δx0 and Δp0. The initial general Gaussian wave packet takes, up to a normalization factor, the form eip0x/ℏe−(1∓iδ)(x−x0)2/4(Δx0)2, where δ=(2Δx0Δp0/ℏ)2−1 denotes a measure of deviation from the minimum uncertainty or the initial position-momentum correlation δ=2Δ(xp)0/ℏ. The travelling Gaussian wave packet takes, up to a time-dependent phase and normalization factor, the form eipcx/ℏe−(1−2iΔ(xp)t/ℏ)(x−xc)2/4(Δxt)2 and the centroid follows the the classical trajectory with xc(t) and pc(t). The position variance is found to have additionally a linearly time-dependent term proportional to δ with both positive and negative signs.
Most physics books do not reveal clearly how the Heisenberg uncertainty principle was derived. This uncertainty comes from the consequence of the wave-particle duality of matter giving statement that position and momentum cannot be measured in the same time. This article tries to reveal mathematics background behind the expression the Heisenberg uncertainty using supported mathematics background such as Fourier transform, Fourier transform integral, the probability of Gaussian distribution and it ends up with the expression of wave function which describe the localized particle giving relation Heisenberg uncertainty principle.
This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
We present several variations of a proof of the position-momentum uncertainty principle that are based on the calculus of variations and does not rely on the Cauchy–Schwartz inequality. We show that the stationary uncertainty wave functions are the Hermite–Gaussian solutions to the quantum harmonic oscillator problem, that the minimum uncertainty wave function is the Gaussian, and that stationary uncertainty wave functions must be their own Fourier transforms. We also provide a calculus of variations proof of the Cauchy–Schwartz inequality. Finally, we discuss the properties of wave functions that are their own Fourier transforms and provide examples of such functions that may be of interest to teachers of undergraduate and graduate level quantum mechanics courses.