Time-dependent Quantum DynamicsTable of contents
Postulate V states how the time-dependent Schrödinger equation describes the evolution of a system’s wavefunction through time.
IntroductionThe Schrödinger equation is the basis for quantum mechanics. The solutions to the equation, known as wave functions, give complete quantum mechanical insight into the system under observation. The Hamiltonian operator, which is specific to the system's environment, acts upon the wavefunction to yield the wavefunction again, accompanied by the energy of the system (the eigenvalue). Postulate V states that the time-dependant Schrödinger equation (1) describes the evolution of a system’s wavefunction through time. \[\hat{H}\Psi_(x,t) = {i}\hbar \dfrac{\partial\Psi}{\partial{t} \;\;\; (1)}\] For a time-independent Hamiltonian Operator, this assumption allows separation of variables: \[\Psi _{(x,t)} =\psi _{(x)} \cdot f_{(t)} \;\;\; (2)\] Equation (2) is substituted into equation (1). \[\hat{H}\psi _{(x)}\cdot f_{(t)} = i\hbar \displaystyle\frac{d\psi _{(x)} \cdot f_{(t)}}{dt} \;\;\; (3) \] Equation (3) is then manipulated so the left side is explicitly in terms of x, while the right side only in terms of t. \[\displaystyle\frac{1}{\psi _{(x)}} {\hat{H}_{(x)}} \psi _{(x)} =\displaystyle\frac{1}{f_{(t)} } i\hbar \displaystyle\frac{df_{(t)} }{dt} \;\;\; (4)\] If both sides (dependent upon different variables) are to be equal, they must both equal the same constant. We shall call this constant E. The left side becomes equivalent to the time-independent Schrödinger equation: \[{\hat{H}} \Psi _{(x)} =E\Psi _{(x)} \;\;\; (5) \] The right side becomes: \[ {\displaystyle\frac{df_{(t)}}{dt}} =\displaystyle\dfrac{-i}{\hbar } E\cdot f_{(t)} \;\;\; (6)\] Equation (6) is manipulated into equation (7) and then integrated: \({\displaystyle\frac{df_{(t)}}{f_{(t)}}} =\displaystyle\dfrac{-i}{\hbar } E\cdot {dt}\) \[\displaystyle\int \frac{df_{(t)} }{f_{(t)} } =\frac{-i}{\hbar } E\cdot \int {dt} \;\;\; (7)\] \[ f_{(t)} =\displaystyle{e^{(-\frac{iE\cdot {t}}{\hbar })}} \;\;\; (8) \] Since \(E=hv=\hbar\omega\), this is also equal to: \[ f_{(t)} =e^{(-i \cdot {\omega} \cdot {t})} \;\;\; (9)\] Equations (9) now plugged into equation (2) give the solution to the time-dependant Schrödinger equation: \[\Psi _{(x,t)} =\psi _{(x)} \cdot e^{(-i\cdot {\omega}\cdot {t})} \; \; \;(10)\] An example of the time dependence of a localized particle in a box can be seen below. First, the constants have to be defined:
Using these constants, the wavefunction and energy solutions can be found: \(E(n)=\dfrac EOF expected: /content/body/div[1]/p[24]/span, line 1, column 2 (click for details) {8\cdot{m}\cdot{a^2}\)
\(\omega{(n)}=\dfrac{2 \pi E_n}{h}\) \psi{(n,x,t)}=\sqrt{\dfrac{2}{a}}{sin\left(\dfrac{n \pi x}{a}\right)}\cdot{e^{-i \omega{(n)}t}} The particle can then be localized by superpositioning the wavefunctions: \(n_{max}=4\) \(n=1,2...n_{max}\) \(\Psi{(x,t)}=\displaystyle\sum_{n}\dfrac{1}{\sqrt{n_{max}}}\cdot\psi{(n,x,t)}\) \(P(x,t)=\Psi{(x,t)}\cdot{\Psi{(x,t)}}\) Using these values and definitions, the following plots of wavefunctions, superposed wavefunctions, and probability distributions can be found (attached below). These plots are captioned to show whether they are real or imaginary and the time.
Real portion of wavefunctions at t=0
Real portion of superposed wavefunctions at t=0
Real portion of probability distribution at t=0
Real portion of wavefunctions at t=1
Real portion of superposed wavefunctions at t=1
Real portion of probability distribution at t=1
Imaginary portion of wavefunctions at t=1
Imaginary portion of superposed wavefunctions at t=1
Real portion of wavefunctions at t=2
Real portion of superposed wavefunctions at t=2
Real portion of probability distribution at t=2
Imaginary portion of wavefunctions at t=2
Imaginary portion of superposed wavefunctions at t=2 Outside linksReferences
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2. For your introductory paragraph I would be more specific as to what exactly Postulate V is. For example I might say "Postulate V states that the time-dependent..." I think this would help with clarification
3. Better labeling of equations might help the reader track as different equations get added together
4. By expanding the webpage (either by inserting in the text or making them questions with answers in the question section) the areas where you say "and Equation.... was manipulated" the reader can better grasp the math involved with solving the equations.
5. Better definition of symbols such as E will help to clarify questions for people who've never heard of this before.
6. Check that you have references and links (such as our textbook or possibly wikipedia)
Overall though the equations look right and it is just detail stuff
1.) Include an introduction (even just 2 sentences) to explain what postulate V is and why it is important.
2.) Visually this page is unappealing - make sure all the text sizes, fonts are the same, think about making equations larger so that all of the terms can be seen and remove extra blank lines. There are a lot of little picky things within the equations that if you have time you should try to fix. (for example there is a question mark in E=hv=?w, hbar instead of the actual symbol)
3.) Your equations are numbered within the text. I would suggest putting a number by the equations so that is easier to figure out which equation is which when you are referring to them.
4.) Explain what a few of the key variables are so that the page is more accessible.
5.) It looks like you spent a lot of time on your figures and they look good - Is there any way you can insert them into the page? For myself I would be highly unlikely to download each of those and actually look at them. Maybe select some of the most important and place them within the page and explain more in detail what each of them mean and then suggest that if a reader is more interested they can look at the additional links.
Also, I might remove a reference to norms for "systems analyzed in this course", though you might discuss "the majority of basic or commonly used quantum mechanical systems/models"
The other commentators already suggested numbers...I agree.
I might add to this statement: "If both sides (dependant upon different variables) are to be equal [IN ALL CASES FOR ALL VARIABLES] -or- [FOR ALL VALUES OF THE DIFFERENT VARIABLES, etc.], they must both..."
Perhaps replace "can be seen below" with "is shown below"
I would consider explaining that the values you chose for your variables in the example are arbitrary and chosen for simplification, even though the definition of "i" is just a definition.
I don't know how far beyond Postulate V you are able to go, but in introducing your topic you may want to include a one-sentence review of what the Schrodinger equation is, and include a chemwiki link to that article.