The probability current density j (r, t) is defined as j(r,...

The probability current density j (r, t) is defined as j(r, t) = ?i / 2m (???* - ?*??) For a one-dimensional wave function of the form ?(x, t) = A exp [i? (x, t)] show that the probability current density can be written as j = ? / m |A|^2 ?? / ?x

Transcript

00:01 Hi there, so for this problem, we are given this wave function, and we are asked to find the probability current density for this wave function.

00:09 So the probability, the current density probability is going to be equal to an h -bar, plams constant, h -bar divided by two times i times the mass, and this times this function, the conjugate of that function, times the derivative of that function with respect to x, and this minus the derivative of this of the conjugate of that function with respect to x and this times the function the wave function.

00:49 Now, let me just write each term separately and then we will introduce it again into this expression.

00:56 So we will have that.

00:57 The conjugate of this function, what we need to do is to simply multiply multiply.

01:16 That by the so we need to multiply each and the terms in here inside the esponemential by minus 1 so we will have that this is a the exponential of since it is negative in this case so it changed to be positive the argument of the exponential and this plus b times the esponential of minus i times k times x now the a and b doesn't change because those are constants.

01:57 Those are real numbers.

01:59 So with that, now we calculate the derivative of this with respect to adds, of the wave function with respect to adds.

02:08 So as you can see, we are going to have, for the first one we are going to have that this is minus a times i times k times the exponential of minus i times k times x.

02:22 And for the other one in here, since it is positive, we obtain b times i times k times the exponential of this.

02:31 We can take out this term in here as a common factor.

02:36 So we will have, well, we have a minus in here, minus a times the exponential of i times k minus minus in here x and this plus b times the s benemtial of i times k times x i forgot the s in here so that's what we have in there now we need to take the derivative part of this expression with respect to x so we will have that that is equal to we will have something similar but it is in the order way around, we will have that that is.

03:19 A is positive, determine positive in this case.

03:24 For the other one, we change this one.

03:30 So that's what we obtained from there.

03:31 And finally, we have the same function.

03:34 We don't need to change that.

03:35 So when we introduce all of this into the expression for the current density, we will have h prime divided by two times the mass, if i'm correct...

Question

The probability current density j (r, t) is defined as j(r, t) = ?i / 2m (???* - ?*??) For a one-dimensional wave function of the form ?(x, t) = A exp [i? (x, t)] show that the probability current density can be written as j = ? / m |A|^2 ?? / ?x

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