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The wave function of particle is given by:w (x) = Acoskx Bsin kxwhere A, B, and k are constants. Show that w is solution of the time-independent Schrodinger equatio...

Question

The wave function of particle is given by:w (x) = Acoskx Bsin kxwhere A, B, and k are constants. Show that w is solution of the time-independent Schrodinger equation with V(x) 0,and find the energy of the particle.

The wave function of particle is given by: w (x) = Acoskx Bsin kx where A, B, and k are constants. Show that w is solution of the time-independent Schrodinger equation with V(x) 0,and find the energy of the particle.



Answers

The wave function of a quantum particle is
$$
\psi(x)=A \cos (k x)+B \sin (k x)
$$
where $A, B,$ and $k$ are constants. Show that $\psi$ is a solution of the Schrödinger equation (Eq. $41.15 ),$ assuming the particle is free $(U=0),$ and find the corresponding the energy $E$ of the particle.

Regular in this question. Every function off a particle, right? This is how we function and you wanna show That's history. Function is a solution to the shorting goes equation which is on the left in the case where the potential is zero ut zero everywhere. Right? This is the situation where the particles of free particle not sure whether it is a solution we were going to do is to substitute inside into this equation. So what do you need used to first find? Why see the second derivative before we function right to a substitute teacher. So simple. Just if you shoot no expression So these are the first derivative we differentiated again, actually get back the we function except that we have an additional negative key square took right So there's or jewelry function wanted my negative Kiska. So what you had to subsequently from here, uh, we put it in into our recreation storing go supervision the negative square to m times negative K square times say, and this issue goes to each bar square key square off to him. I'm sigh and this is indeed equals two a constant, some constant multiplied by citing the re function and this is t say Masti right hand side where the right hand side shows that we might play energy e which is a constant 50 way function. So it means that this we function actually satisfy ice de shorting ghost equation and it's vetted for all X in the case. Where are E? How energy iss this particle constant over here, Boss square key square for do it.

The Schrodinger equation is given by negative H Bar squared over two n times. A second derivative of SAI with respect to eggs and in this case is equal to the Energy times. Sigh because you is equal to zero, so we'll go ahead and get first derivative, and then we'll get a second derivative so deci with respect to X First derivative is negative k times a sign que x because the first derivative of a co sign is a negative sign. And similarly, the first derivative of the sign is a co sign. So here we have K tires. Be Times Co sign of K X. All right, so now we'll get the second derivative, and that is the derivative of the first derivative. So here we'll just write d of d d x of di sai d x, and we can write. The second derivative out here is equaling this. So now we have a second kay that comes out times k A. And the derivative sign is co sign. So we have co sign K X and the derivative of co sign. It's negative signs. So now we have a negative K that comes out times k be and sign K X, and this could be simplified as negative K squared a co sign. Okay, X minus case. Where'd being sign Que eggs And we can simplify one more time by factoring out negative k squared from both of these terms. So we have negative K squared times a co sign. Okay, X plus B sign. Okay, X Now, if you notice this is really just the original function wave function times negative k squared. So we'll just write this for short is negative. K square times sigh. Now we can substitute this back into the schrodinger equation. We have negative h bar squared over to em times the second derivative which we just obtained. And we can write this now as negative h bar squared were to in times negative case where Sai Equalling e Sigh, which is the other side of the Schrodinger equation. And this will now simplify to h bar Squared K squared Sigh and we have two in the denominator Equalling e sai. And indeed, we have shown that this solution works basically a co sign k x plus b sine K X is a solution of the Schrodinger equation because we get this function on both sides of the equation times a constant and in this case, this constant in this constant equal each other and we get the relation E equals H bar squared case where over to em And that's the energy for a free particle.

Okay, So first of all in this problem Ah, the suri. The time independent equation for a free particle is describe us. No, not TV. Age square divided by two m off second, partial derivative. With respect of the position off these wave function that is equal the energy off this particle times the way function. Okay, So since we know what is the way function of this problem, we just need to make the second derivative off this way function. So as we can see, the second derivative is which here, off the ray function, which is a a sign off K x will us be course sign off K X. This is going to be cruel. It's which here, the second partial derivative is going to be, um, negative que square Because the first derivative of sign school sign the second derivative off we'll sign is negative sign. Therefore, this is negative. Hey, sign off K x. Ah, the loss. Actually negative. The K square may gross sign off K X because the second derivative of consign is negative co sign And since we have to derivatives, we have two case and here Okay, square. Therefore, as we can see the left side off. This equation is going to be. It's Putin here. The left side of the short integration is negative. H divided by two m que square was a teeth because we have a negative in the year that multiplies their mood supplies a sign off K X lost me course Sign off K X that is equal the right side the right side We have the energy times, the wave function site X and the way function beside off X is precisely this solution And here therefore this is equal age is square Hey, is square to em That is equal energy. Therefore, not only we can see from here that the way function the problem gives us it is a solution for the time independent shorten his aggression. But we can also see that the value off energy is described by this relation here. And remember, h k is the momentum. Therefore, the energy is actually the energy of a free particle in the time Depend time independent. Sure, integration is just P square divided by two m and we know this result by the book it is correct. Ah, I think this is it. We should. In this problem, we should show that that way function is a solution. This is proof here, okay? And we have to find it Energy off this of this rib article. Wow. The energy off this free particle is in here, that's all. Let's define our answer to this problem. Thanks for watching.

In this exercise, we have to show that the wave function sigh of acts shown here in the screen. So I have actually go to a times e to the I. K. X is a solution to showing this equation. This is the second equation here that has a constant potential energy. You zero there you zero is smaller than the total energy of the particle eat. We have to find on expression for K that solves this equation. So basically, I'm going to take the left hand side of this equation and apply on our way function. So we have minus h bar squared, divided by two women times the secondary in space off aid to times E to the I k X plus you zero times a to the I Thio I'm sorry eight times e to the I k X. This is equal to minus the bar square developed by to win the second derivative off the complex exponential is equal to, uh minus k squared times e to the i k x 10 a eat to the k x this'll se's h bar squared case squared, divided by two whim plus you zero times a times e to the i k x This is H bar squared Key square developed by to swim Plus you zero times sigh. Yeah, And for this to be a solution to shorten his equation, we want this to be able to the energy Times side. And this is true if h bar squared k squared available to em was you zero is equal to eat. Yeah. Okay, so in this case, K must be shoe the square root of 2 a.m. Times e minus you zero the value by H bar. Okay, so I've shown that are proposed wave function as a solution If K has this value here, uh, furthermore, we also have to find what is the They're really wave ling. Actually, we have to relate k to the Beverly Waveland. So remember that key The momentum of the of the particle is equal to H divided by the wavelength The liberal you able in London Also remember that the kinetic energy All right, look kinetic energy K is equal to p squared divided by two m So this is equal to age over Lunda squared divided by two n. This is the kinetic energy of the particles. Uh, so basically, wow, notice that the kinetic energy can also be written as h bar squared, K squared, divided by two. Mm. Okay. This is just, uh we can obtain this by remembering that the kinetic energy kim redness minus h bar square, development, whim, times the secondary in time. Offside. So and this must return as the kinetic energy types sigh. So I'm gonna take these two equations here can make them equal. So we have h bar square K squared, divided by two n is equal to age over London squared, divided by two web. So the two m's cancel out. I'm going to rewrite H bar squared as age divided by two pi square to have h squared about a four ply squared. Okay. Square is equal to each square. The very well in the square. The ages console out. Get that K is equal to two pi over London and this concludes our exercise. Okay?


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