Prove that the probability density and the probability...

Prove that the probability density and the probability current density at position $\mathbf{r}_0$ can be expressed in terms of the operators $\mathbf{r}$ and $\mathbf{p}$ as expectation values of the operators $$ \rho\left(\mathbf{r}_0\right) \rightarrow \delta\left(\mathbf{r}-\mathbf{r}_0\right) \quad \mathbf{j}\left(\mathbf{r}_0\right) \rightarrow \frac{1}{2 m}\left[\mathbf{p} \delta\left(\mathbf{r}-\mathbf{r}_0\right)+\delta\left(\mathbf{r}-\mathbf{r}_0\right) \mathbf{p}\right] $$ Derive expressions for these densities in the momentum representation.

Prove that the probability density and the probability current density at position $\mathbf{r}_0$ can be expressed in terms of the operators $\mathbf{r}$ and $\mathbf{p}$ as expectation values of the operators
$$
\rho\left(\mathbf{r}_0\right) \rightarrow \delta\left(\mathbf{r}-\mathbf{r}_0\right) \quad \mathbf{j}\left(\mathbf{r}_0\right) \rightarrow \frac{1}{2 m}\left[\mathbf{p} \delta\left(\mathbf{r}-\mathbf{r}_0\right)+\delta\left(\mathbf{r}-\mathbf{r}_0\right) \mathbf{p}\right]
$$

Derive expressions for these densities in the momentum representation.

Key Concepts

Operator Expectation Values

In quantum mechanics, physical quantities are represented by operators, and the measurable values of these quantities are found by taking expectation values. This concept underpins how the probability density and current density are calculated as averages over the quantum state, providing a bridge between the abstract operator formalism and observable predictions.

Wavefunction Representations

Quantum states can be expressed in different bases, notably the position and momentum representations. The wavefunction in a given representation contains all the information about the state and allows one to compute probabilities and expectation values for observables associated with that basis.

Fourier Transform

The Fourier transform is the mathematical tool used to convert between the position and momentum representations of a quantum state. This transformation is essential for deriving the expressions of operators such as the delta function and momentum operator in the momentum representation.

Probability Density

The probability density gives the likelihood of finding a quantum particle at a particular position. In the operator formalism, this density is expressed as the expectation value of a position-space delta function operator, which localizes the measurement at a specific point.

Probability Current Density

The probability current density quantifies the flow of probability in space and is related to the conservation of probability. Its operator form, involving the symmetrized product of momentum and a position-localized delta function, ensures that the continuity equation is satisfied in quantum mechanics.

Momentum Representation

The momentum representation is a formulation of quantum mechanics where states and operators are expressed as functions of momentum. This representation is particularly useful for problems involving translational symmetry or when momentum-related properties are of interest, and it requires reformulating position-space operators via Fourier transforms.

Symmetrization of Operators

In quantum mechanics, the order of non-commuting operators matters. Symmetrization is the process of averaging over different orderings to ensure that the operator is Hermitian, which is necessary for representing observable quantities. In the context of the probability current density, this leads to a symmetric combination of momentum and the delta function operator.

Transcript

00:01 In this question, we want to show that a physical state is specified when the density function and the probability current is specified.

00:16 Okay, so there are three parts in this question.

00:23 In part a, we are given.

00:29 The state wave function is given to be a square root of row r, okay, e to the i, cr, okay.

00:42 So the c is called the argument, okay.

00:46 And then we want to show the probability current function.

00:54 So the definition of probability current.

01:08 Okay.

01:09 So this is j .r is equal to h bar over 2im, side star gradient of si minus si r gradient of si star gradient of si star okay so um so gradient of si r is the first function we need to do so look at look at this function so we have two functions of position so the we need to do product rule okay so i'm going to differentiate the part with the row first okay to not write the gradient of row times e to the i c kr and then plus square root of row so then if you multiply side star, okay, to the gradient of psi, and so you get a square row into the negative i -c times 1 over 2 root row, gradient of row into the i -c.

02:53 So now we have a lot of algebra to do in this question.

02:57 So we just embrace it and get through it.

03:10 So you can see that the this turns will kill each other.

03:15 Okay, because they are complex conjugate of each other and the first time you get half.

03:21 Gradient of row plus i row gradient of kher.

03:29 Then we do the second term.

03:31 So gradient of the side star...

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