Consider a particle moving in one dimension, which we shall...

Consider a particle moving in one dimension, which we shall call the $x$-axis. (a) What does it mean for the wave function of this particle to be $normalized$? (b) Is the wave function $\psi(x) = e^{ax}$ , where a is a positive real number, normalized? Could this be a valid wave function? (c) If the particle described by the wave function $\psi(x) = Ae^{-bx}$, where $A$ and $b$ are positive real numbers, is confined to the range $x \geq 0$, determine A (including its units) so that the wave function is normalized.

Transcript

00:01 This question is stated that consider a particle moving in one direction which we shall call the x -axis so suppose a particle is moving in the x direction and this is the state of this particle and the state of this particle is represented by some wave function that is x x and the the first question in the what does it mean for the wave function of this particle is to be normalized that means we have to explain what is the meaning of the normalization of this way function when a particle is moving in this direction and the state of this particle can be presented by wave function the state of particle sine x is equal to e power i k x where k is the momentum wave vector and x and this x representing the direction here k is 2 pi by lambda and lambda is the wavelength of the this and k is normally known as the momentum wave vector so here we have to explain there what is the meaning of this normalized so if a wave function is normalized if a wave function is normalized then it should follow it should satisfy the normalization condition which is minus infinitive to plus infinitive si complex conjugate x and the x x x and d x must be one and the meaning of this that in the probability of finding the particle probability of finding the particle must be one that is maximum probability of that means particle is present in some regions suppose we want to calculate on to know the particle in this region then the probability of finding the particle in this range is one so this if any wayfunction we satisfy this condition that way function is known as the normalized wayfunction so here this is is x is equal to e power i k x so if you find know this wayfunction in normalize or not then we have to apply this condition and here x that is complex conjugate is equal to e power minus i k x and now apply this condition that is si x and si x d x minus infinity is equal to this is e power minus i k x and this is i k x and this is i k x and this is i kx and d x and this value is cancelled out that means the wave function minus infinity to plus infinite that is d x that means x and this is minus infinity to plus infinity that means the wave function undefined that means this is not one the given way function is not nominalized that it is not equal to 1 hence this wave function the function is not normalized and one more condition for this the way function psi x must be 0 at x is equal to plus minus infinite now another part of this question is is the wave function is the wave function x x is equal to e power a x where is a positive real number is normalized here this is the b part and the wave function is given x is equal to e power a x to check the normalization of this bay function we have to apply this condition minus infinitive to plus infinity x x complex conjugate of this function and the si x and then d x this is minus infinity to plus infinity e power that it is a real way function that means e power 2 a x and d x now we have to find the integration of this one so this is e power 2 a x even by 2a and limits are minus infinity to plus infinity to that is 1 upon 2a e power infinity minus e power minus infinity and this value is 0 that means 1 by 2a and e power infinitive again this value is not 1 so hence x is equal to e power a x is not normalized now could this be a valid way function? next part of this question, we have to check could you, could this way function is a valid way function? answer is no.

08:40 Because, si x is equal to e power a x.

08:50 When you see this x is equal to infinite, this value is not zero and x x is equal to minus infinite this is zero so this way function will be valid if the wave function at x is at very large distance the value of this way function is zero but as you can see when you are increasing the value of the x this wave function is below up if you plot this is x so here this when x is equal to 0 this value is around 1 and now when you increase the x this wave function is increasing increasing increasing so and the way function which is valid should not blow up it can blow up to a particular limit but after a at a very large distance it should be zero way suppose the way function at this is x is equal to infinite then it should be zero but here this way function is increasing increasing hence this way function is not a valid way function and the c part of this question if the particle described by the way function si x is equal to a e power minus bx this is c part psi x is equal to a e power minus b x then we have to here capital a and the b are positive real number is confined to the x g greater and equal to zero determine the value of the a so that the way function is normalized.

11:33 So we have to find out the value of this a.

11:37 And here, x is this value...

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