So, after some rearranging, we have. Equation 3. Fortunately, it has already been solved, and so we will simply examine the solutions. But first, it is important to note that the wavefunction can be described as the product of three distinct functions. A more in-depth analysis of Equation 3. We will just make use of the result where. By restricting the particle to portential wells, we are able to derive the particle's wavefunction--both inside and outside of the well--and the quantum mechanical energy that the particle possesses due to its wave-like behavior.
Calculate the ground state energy of a baseball that weighs grams that is in a football field yards long. Note: It is important to realize that in the case of the baseball in a football field, the quantum mechanical energies associated with its wavelike behavior are so small that they are essetially negligible. This is why we see a continuum of energy associated with its motion, whereas the electron has defined quanta of energy in which it can orbit.
As Figure P. This assessment only works for symmetrical wavefunction distributions. For all others, Equation P. The Particle in a One Dimensional Box The Infinite Well The particle in a box is a very simple model developed in order to understand the behavior of a particle i.
Figure 1: depiction of a particle a and wave b confined to a well with barriers of infinite potential energy. Particle in a Finite Well What distinguishes the Finite Well from the Particle in a Box scenarios are that for the finite well, the potential energy, V, of the barriers do not approach infinity. Figure 2: Depiction of the wave behavior of a particle with energy, E, in a Finite Well with a finite potential energy, V, equal to a constant, P.
The Hydrogen Atom The solutions to the Infinite Well and Finite Well are useful for describing the behavior of a particle when confined to a small region of space with large and small potential barriers, respectively.
Problems 1. Such a reason may not exist. However, we use mathematics to model the universe all the time e. It is a number that can be used within a theoretical framework that is tested with experiment.
If the predictions are consistent with experiment, we label the theory's predictions correct. The wave-function is not real; it does not represent any physical phenomenon. For example, to find an expectation value of a particular wave-function, we typically find. In differential equations, imaginary coefficients on first order linear equations change exponential behavior into sinusoidal behavior.
The complex numbers should be thought of more as a two dimensional vector space over the real numbers rather than as some sort of mystical imaginary thing. Complex numbers are especially good at simplifying two dimensional equations when there is some sort of symmetry involved, which we have here. Finally, the imaginary factor corresponds to a 90 degree rotation of some sort.
This is non-rigorous because the Hamiltonian can alter the phase on the right side. Its well known QM behaves strangely. This arises from the super-position of the wave functions. The amplitude of the wave-function is complex. The square of the amplitude is probability. Another way of putting this is that the weirdness of QM comes from taking the square root of probability.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Since the Dirac equation is first order in space and time, and the Klein-Gordon equation and classical wave equation are second order in space ant time, quite a different picture would emerge. Starting with a real solution, which is a plane exponential growth function for 22 we then have from Eq. Note that does not satisfy Eq. All quantities k 2 , 2 , 2 are real as is X and i.
The form of the solution in Eq. He find that solutions in an imaginary spacetime geometry yield growth equations. Equation 22 is a linear form. We also have another solution in Eq. Let us briefly discuss the introduction of a non-linear term with a small coupling constant. We introduce a "potential" term which is coupled by a small coupling constant, g 2 , and is associated with an attractive force. If the coupling term is small, then solutions can be determined in teems of a perturbation expansion.
This field satisfies the Dirac equation and introduces an additional term in the Lagrangian. In Ref. In the most general case we have functional dependences X x,t and i x,t. With the quantum superposition principle, we can combine real and imaginary parts. For the x-directional form of Eq. We discuss this assumption and tachyonic implications in Ref. Equation 32 is defined on a four-dimensional space x,X,t, i. In the first approximation, we will choose so that we have 34 Motivation of this approximation can be found in our discussion of remote connectedness properties, diagrammed in Figures 2 and 3 of the previous section.
Let us rewrite Eq. From examination of the forms of Eq. This result is similar to that from the more comprehensive field theoretic argument for the Dirac equation.
The associated metric space for X,t, i defines a remote connectedness geometry.
0コメント