How to of the Day

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• How to Solve the Particle in a Box Problem
• How to Fringe a Shirt
• How to Reject a Guy Who Wants Your Number

How to Solve the Particle in a Box Problem Posted: 14 Jul 2017 05:00 PM PDT In quantum mechanics, the particle in a box is a conceptually simple problem in position space that illustrates the quantum nature of particles by only allowing discrete values of energy. In this problem, we start from the Schrödinger equation, find the energy eigenvalues, and proceed to impose normalization conditions to derive the eigenfunctions associated with those energy levels. EditSteps Begin with the time-independent Schrödinger equation. The Schrödinger equation is one of the fundamental equations in quantum mechanics that describes how quantum states evolve in time. The time-independent equation is an eigenvalue equation, and thus, only certain eigenvalues of energy exist as solutions. ${\displaystyle {\hat {H}}|\psi (x)\rangle =E|\psi (x)\rangle }$ Substitute the Hamiltonian of a free particle into the Schrödinger equation. In the one-dimensional particle in a box scenario, the Hamiltonian is given by the following expression. This is familiar from classical mechanics as the sum of the kinetic and potential energies, but in quantum mechanics, we assume that position and momentum are operators. ${\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x)}$ In position space, the momentum operator is given by ${\displaystyle {\hat {p}}=-i\hbar {\frac {\mathrm {d} }{{\mathrm {d} }x}}.}$ ${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {{\mathrm {d} }^{2}}{{\mathrm {d} }x^{2}}}+V(x)}$ Meanwhile, we let ${\displaystyle V(x)=0}$ inside the box and ${\displaystyle V(x)=\infty }$ everywhere else. Because ${\displaystyle V(x)=0}$ in the region that we are interested in, we may now write this equation as a linear differential equation with constant coefficients. ${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {{\mathrm {d} }^{2}\psi }{{\mathrm {d} }x^{2}}}=E\psi }$ Rearranging terms and defining a constant ${\displaystyle k^{2}={\frac {2mE}{\hbar ^{2}}},}$ we arrive at the following equation. ${\displaystyle {\frac {{\mathrm {d} }^{2}\psi }{{\mathrm {d} }x^{2}}}+k^{2}\psi =0}$ Solve the above equation. This equation is familiar from classical mechanics as the equation describing simple harmonic motion. The theory of differential equations tells us that the general solution to the above equation is of the following form, where ${\displaystyle A}$ and ${\displaystyle B}$ are arbitrary complex constants and ${\displaystyle L}$ is the width of the box. We are choosing coordinates such that one end of the box lies at ${\displaystyle x=0}$ for simplicity of calculations. ${\displaystyle \psi (x)=A\sin kx+B\cos kx,\quad 0<x<L}$ Of course, the solution is valid only up to an overall phase, which does change with time, but phase changes do not affect any of our observables, including energy. Therefore, for our purposes, we will write the wavefunction as only varying with position ${\displaystyle \psi (x),}$ hence the usage of the time-independent Schrödinger equation. Impose boundary conditions. Remember that ${\displaystyle V(x)=\infty }$ everywhere outside the box, so the wavefunction must vanish at the ends. ${\displaystyle \psi (0)=A\sin(k\cdot 0)+B\cos(k\cdot 0)=0}$ ${\displaystyle \psi (L)=A\sin(kL)+B\cos(kL)=0}$ This is a system of linear equations, so we may write this system in matrix form. ${\displaystyle {\begin{pmatrix}0&1\\\sin kL&\cos kL\end{pmatrix}}{\begin{pmatrix}A\\B\end{pmatrix}}=0}$ Take the determinant of the matrix and evaluate. In order for the above homogeneous equation to have nontrivial solutions, the determinant must vanish. This is a standard result from linear algebra. If you are not familiar with this background, you may treat this as a theorem. The sine function is 0 only when its argument is an integer multiple of ${\displaystyle \pi .}$ ${\displaystyle {\begin{aligned}-\sin kL&=0\\kL&=n\pi ,\ n\in {\mathbb {Z} }\end{aligned}}}$ Recall that ${\displaystyle k={\sqrt {\frac {2mE}{\hbar ^{2}}}}.}$ We may then solve for ${\displaystyle E.}$ ${\displaystyle kL={\sqrt {\frac {2mE_{n}}{\hbar ^{2}}}}L=n\pi }$ ${\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}n^{2}}$ These are the energy eigenvalues of the particle in a box. Because ${\displaystyle n}$ is an integer, the energy of this system can only take on discrete values. This is a chiefly quantum mechanical phenomenon, quite unlike classical mechanics, where a particle can take on continuous values for its energy. The energy of the particle can only take on positive values, even at rest. The ground-state energy ${\displaystyle E_{1}}$ is called the zero-point energy of the particle. The energy corresponding to ${\displaystyle n=0}$ is not allowed because this physically represents that no particle is in the box. Because the energies increase quadratically, higher energy levels are spread out more than lower energy levels. We will now proceed to derive the energy eigenfunctions. Write out the wavefunction with the unknown constant. We know from the constraint of the wavefunction at ${\displaystyle x=0}$ that ${\displaystyle B=0}$ (see the first equation in step 4). Therefore, the wavefunction will only contain one term from the general solution of the differential equation. Below, we substitute ${\displaystyle k={\frac {n\pi }{L}}.}$ ${\displaystyle \psi _{n}(x)=A\sin {\frac {n\pi x}{L}},\ n\in {\mathbb {Z} }}$ Normalize the wavefunction. Normalizing will determine the constant ${\displaystyle A}$ and will ensure that the probability of finding the particle in the box is 1. Since ${\displaystyle n}$ can only be an integer, it is convenient to set ${\displaystyle n=1}$ here, as the only purpose of substituting a value is to obtain an expression for ${\displaystyle A.}$ It is helpful to know the integral ${\displaystyle \int _{0}^{\pi }\sin ^{2}x{\mathrm {d} }x={\frac {\pi }{2}}}$ when normalizing. ${\displaystyle 1=\int \psi ^{*}(x)\psi (x){\mathrm {d} }x=\int _{0}^{L}A^{*}A\sin ^{2}{\frac {n\pi x}{L}},\ n\in {\mathbb {Z} }}$ ${\displaystyle {\begin{aligned}{\frac {1}{|A|^{2}}}&=\int _{0}^{L}\sin ^{2}{\frac {\pi x}{L}}{\mathrm {d} }x,\ u={\frac {\pi x}{L}}\\&=\int _{0}^{\pi }{\frac {L}{\pi }}\sin ^{2}u{\mathrm {d} }u\\&={\frac {L}{\pi }}{\frac {\pi }{2}}\end{aligned}}}$ ${\displaystyle A={\sqrt {\frac {2}{L}}}}$ Arrive at the wavefunction. This is the description of a particle inside a box, surrounded by infinite potential energy walls. While ${\displaystyle n}$ can take on a negative value, the result would simply negate the wavefunction and result in a phase change, not an entirely different state. We can clearly see why only discrete energies are allowed here, because the box only allows those wavefunctions with nodes at ${\displaystyle x=0}$ and ${\displaystyle x=L.}$ ${\displaystyle \psi _{n}(x)={\sqrt {\frac {2}{L}}}\sin {\frac {n\pi x}{L}},\ 0<x<L}$ EditTips When normalizing, substituting an appropriate integer for ${\displaystyle n}$ and performing the resulting u-substitution will always return the correct answer for ${\displaystyle A,}$ since the change in the derivative is compensated by the change in the boundary. Verify this by setting ${\displaystyle n=2,}$ or any other positive integer, and normalizing again.

How to Fringe a Shirt Posted: 14 Jul 2017 09:00 AM PDT Fringing a shirt is an easy and fun way to produce a totally new look from shirts you already have in your wardrobe. There are several ways to decorate a fringed shirt, and you can experiment with different decoration methods to create a unique look all your own. EditSteps EditMaking The Fringe Shirt Gather your materials. To create your fringe shirt you will need a few materials: A t-shirt (men's t-shirt provide a looser fit, women's t-shirts provide a tighter fit)[1] Sharp scissors (fabric scissors work best) Ruler Chalk or fabric marker Decorative beads[2] (optional) Mark where the fringe will start. Put on the t-shirt. Stand in front of a mirror and use the chalk to make a provisional line across the front of your shirt where you want the fringe to start.[3] Keep in mind that depending on how high you cut the fringe, you may be exposing your stomach. Some people like this look, while others prefer for the fringe to start around the waist of their pants or shorts. Measure the chalk line. Lay the shirt down on a flat surface such as the floor or a table. Measure the distance from each armpit of the shirt to each end of your drawn chalk line.[4] This will help determine if your line is straight and even on both sides. If the two measurements are different in length, measure again and mark the place needed to make the chalk line straight and even. Then lay your ruler across the shirt and redraw the horizontal chalk line, connecting the two even measurements. For example, if the chalk line to the left side armpit measures , and the chalk line to the right side armpit measures , determine which length you rather have. Measure and mark the corresponding length on the side of the shirt you're adjusting. Once both ends of the chalk line are equidistant from the armpits, connect the two markings to even out your chalk line.[5] This line will be where you stop cutting the strips of fringe. Mark your fringe lines. Lay your ruler on the drawn chalk line and use your chalk to mark ½ inch intervals where your strips will be cut.[6] Once you finish marking the top of the shirt, lay your ruler on the bottom of the shirt and again, mark out ½ inch intervals where you are going to cut your fringe strips. Then, lay your ruler vertically on the shirt, and connect the ½ inch marks.[7] This will give a clear line on which to cut your fringe. You could also free cut your strips without marking lines to cut along, but cutting along measured lines will make your shirt look less sloppy.[8] If you plan to thread beads onto the fringe, thinner strips will allow the beads to slide onto the fringe strips more easily. Cut your shirt. Cut the bottom hem off of your shirt, right above the top stitching. Use your scissors to cut along the vertical ½ lines you measured for your fringe. You can cut both sides of the shirt at once.[9] Just be sure that both the front and back of the shirt are spread out and flat when you start cutting. Stop cutting once you get to the top chalk line. The first and last strip you cut will be on the sides of the shirt. That means that the side strips will actually be an inch thick since it's a ½ inch on the front of the shirt connected to ½ inch on the back of the shirt. Cut these thicker strips in half down the middle, to make them the same as the rest of your fringe strips.[10] Stretch out the strips. When all the fringe strips are cut, use one hand to hold the t-shirt down and in place. Use your other hand to tug on the bottom of the strips, to curl in the cut sides and make the strips look more like fringe.[11] You can leave your shirt as it is for a basic fringe top, or you can further embellish your shirt even more unique look. EditDecorating The Fringe Shirt Knot the fringe. Take two neighboring fringe strips, and knot them about an inch down from where the strips begin.[12] Do this for all the strips on your shirt. You can leave your shirt like this with small knots going around your shirt, or you can add another layer of knots to give your shirt a crisscross knotted look.[13] Crisscross knot the shirt. Once you have small knots going all the way around your shirt, knot together a right strip from one knot, with a left strip from an adjacent knot, and knot those two strips together about an inch down from the original knots.[14] Keep knotting together outside strips from adjacent knot pairs, to create a crisscross effect in your shirt.[15] Stagger the fringe. Stagger the fringe by simply cutting some of the fringe strips to make them shorter than others.[16] You can try cutting every other strip shorter, only the strips in the back of the shirt, or random strips. Bead the fringe. Slide decorative beads onto the strips of fringe. You can add as many beads as you want, but most fringe looks best with about 1-3 beads on a strip.[17] Once you're finished sliding beads onto a strip, make a small knot at the bottom of the strip to keep the beads in place. You can add beads to a plain fringed shirt, a shirt with crisscross knots, and staggered fringe shirts. It's all about personal preference. Experiment to see which look you like best. EditTips If you'd like thinner or thicker strips, go for it. If you're cutting very thin strips, it's recommended that you cut the t-shirt into thirds first, then cut those thirds into much smaller strips. This makes handling the t-shirt much easier than trying to cut tiny strips across the whole t-shirt at once. To really personalize the t-shirt, consider tie-dyeing, fabric painting or embroidering it as well as creating the fringe. Fringe shirts make a great market stall or fundraising sales item for school fairs. Practice on a thrift store shirt the first time if you're worried about not getting it right. Once you're confident, you can proceed to a t-shirt that you really want to look fantastic! EditRelated wikiHows Cut a T Shirt Make a Slit Sided T Shirt EditSources and Citations https://www.youtube.com/watch?v=sfnawM6ZE6E Cite error: <ref> tags exist, but no <references/> tag was found

How to Reject a Guy Who Wants Your Number Posted: 14 Jul 2017 01:00 AM PDT If a guy is interested in you and asks for your number, but you really don't like or have an interest in him, the best thing is to get to the point and say "no thanks." Of course, that may not be your style, or you may feel safer trying a different option. If so, move on to using decoys and sidestepping the issue. EditSteps EditBeing Direct Just tell him no. You don't necessarily have to give him an explanation or a reason why. You can just tell him that you're not interested, or you don't want to give him your number. If you use this method, make sure you really don't want to keep in touch with the guy, even if you see him again.[1] For instance, you could say, "Thanks for asking, but I'm not interested." You can also just give him a number to stay in touch as friends, as long as you are abundantly clear about it. For instance, you could say, "I don't mind giving you my number, but I would only do so as friends. I'm not interested in anything else right now." Try a little less direct approach. If you want, you can take a direct, but not quite as direct approach. For instance, you can try to soften the blow by leading with something you like about the guy. That way, the rejection won't come as hard.[2] As an example, you could say, "You're really attractive, but I don't think I'm in a place to be dating right now. I have to say no." This approach helps make it less about him, so he won't feel as hurt. Make sure you include the word "no." When you're using the direct or indirect approach, "no" needs to be in what you say. If you hem and haw around it without actually using "no," you leave him thinking he may still have a chance with you. You don't need to be mean, just direct.[3] For instance, saying "I'm not sure I want to date right now" leaves it a little uncertain. Instead, make sure you make it clear. "I'm not interested in dating right now, so I have to say no." Be polite but firm. Thank the guy for his interest. Say that you're flattered he is interested. However, let him know that you're not looking for a relationship at the moment, and hence, you're not able or willing to share your number. Don't apologize. It's a natural reaction. You want to apologize for making the person feel bad. However, you don't need to apologize for not wanting to give out your number. In addition, it can draw attention to the fact that you feel sorry for him, which in turn can make him feel worse.[4] EditUsing Decoys Give out a fake number. This option can be a little risky, since he can call it right then and see if it is actually your number. However, it can be a way to get someone to stop asking you out in an indirect way.[5] Also, make sure the number you're giving isn't someone else's number. You can look up most numbers online to see if they connect to a person. You don't want to be giving out a person's number to a stranger. If you happen to run into him again and he asks you about it, just ask to see the number, then say "Whoops! Wow, I spaced out. I totally wrote down one of the numbers wrong." And when he asks for the number again you can do the same thing. (If you're going to see him more than twice, it's best to just tell the truth.) Give out a hotline number. Some hotlines have been designed specifically for rejecting guys. Others are just a silly way to give a fake number. Either way, you can use these numbers to hand out to the guys you meet so you don't have to give your number.[6] Another option is the rejection line. It explains to the guy that you've given him a false number as a way of rejecting him with a bit of humor thrown in. The number is (605) 475-6968.[7] Use a friend as a boyfriend. Another option is to take a guy friend with you when you're out who's willing to pretend to be your boyfriend. That way, if you need to reject someone, you can use him as a decoy. You could say, "Oh, I would if I were single, but I'm here with my boyfriend." It helps to grab your decoy boyfriend's arm at this point. EditSidestepping the Issue Use safety as an excuse. One way to sidestep the issue is to say that you don't ever give out personal information to strangers as a way of staying safe. Of course, this has an element of truth for most people, so you're really not lying.[8] You could say, "I'm sorry, I don't give out my number to strangers. I've had some bad experience with doing that in the past, so I don't give out my number anymore." If you make it so he realizes it's a general rule, he won't feel so personally rejected. Use a boyfriend as an excuse. Even if you don't have a boyfriend, you could use a pretend boyfriend to make the rejection easier. If the guy thinks you have a boyfriend, then it's nothing personal against him.[9] As an example, you could say, "I can't give you my number. I have a boyfriend." Most guys will walk away without another thought at that point. Turn the tables. Another way to sidestep the issue is to ask for his number instead. That way, the power is in your hands, and you can do with it what you want. Just make sure to take the number yourself. Don't hand him your phone, as he could use it to give himself your number.[10] It may seem cruel to take a number just to blow the guy off later. However, if a guy won't leave you alone, it's a good way to get him to stop. Stay safe. Don't be afraid to walk away if you need to do so. If a guy turns hostile after rejection, get security involved and then leave where you are as soon as possible. It's always better to be safe than sorry.[11] If you're not in a place that has security, tell management or call the police if you feel your safety is threatened. It's also a good idea to go out in groups. While guys don't limit asking girls out to clubs, when you are going out to the club, take some friends with you. EditWarnings If he wants you to meet him somewhere you've never been before, asks you to ride in his car alone with him, or wants to meet where no one can see you, take extreme caution. Let someone else know where you are, bring someone along with you without his knowledge, change the location, or just simply refuse to meet with him alone. EditRelated wikiHows Get a Guy to Stop Liking You After You Lead Him On Be Friends with a Girl That Rejected You Make a Guy Ask You for Your Phone Number Meet Guys Give a Guy an Answer when He Asks You Out EditSources and Citations Cite error: <ref> tags exist, but no <references/> tag was found

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