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I wanted to try to understand density operators better so I thought to try to actually solve a simple problem with them. I started with the Neumann equation

$$\frac{d\rho}{dt}=\frac{1}{i\hbar}[H,\rho]$$

From which I got the time-independent one

$$[H,\rho]=E\rho$$

I defined the potential

$$V(x)=0 \text{ for}, x \geq 0, x \leq a$$ $$V(x)=\infty \text{ for}, x<0, x>a$$

I defined the density operator like this because I want to get the probabilities of seeing the particle in a space x. I am not really sure how to define $\rho$ for a continuous basis. I never saw someone using an integral for $\rho$ so I keep it as a sum. $$\rho_x=\sum_{x=0}^a p_x|x\rangle\langle x|$$

After I did some calculation I got stuck here

$$\frac{d^2\rho_x}{dx^2}+\alpha^2\rho_x=0$$ $$\alpha^2=\frac{2mE}{\hbar^2}$$

The above equation I rewriten to solve it and I got it to this form, and my question is how exactly do I continue from here? $$\lambda^2+\alpha^2=0$$

If I was to try to solve this normally I will get $$\rho_x=Ae^{i\lambda x}+Be^{-i\lambda x}$$

But I don't think that I should get the exact form as if I was using the wavefunction. How to I get a result in a matrix form? Since the density operator is a matrix.

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    $\begingroup$ You can solve for wave functions and assemble the density matrix from the wave functions. $\endgroup$
    – Roger V.
    Nov 29, 2022 at 10:08

1 Answer 1

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TL;DR

I think it's easier to work in the eigenbasis of the Hamiltonian and do the solution there, which I will show below. If you want the position basis, you need to expand the density operator as

$$ \hat{\rho}(t) = \int_{-\infty}^{\infty} \mathrm{d}x \int_{-\infty}^{\infty} \mathrm{d} x' \rho(x,x',t) | x \rangle \langle x' |, $$

substitute it into the von Neumann equation and then you end up with

$$ i \hbar \frac{\partial}{\partial t} \rho(x,x',t) = \left[ -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right] \rho(x,x',t) - \left[ -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x'^2} + V(x') \right] \rho(x,x',t).$$

The form of this equation is suggestive of a separation of variables $\rho(x,x',t) = c(t)\psi(x)\phi(x')$. It turns out that it indeed works and you will just get back the stationary Schrödinger equation for both $\psi(x)$ and $\phi(x')$:

$$ \left[ -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right] \psi(x) = E \psi(x).$$ (Actually, you get the complex conjugate equation for $\phi(x)$ but the eigenfunctions of this problem are real.) You can now find the spectrum $E_n,~n=1,2,\ldots$ and the eigenfunctions $\psi_n(x)$.

Putting it all together, write the density matrix elements in position basis in series form

$$ \rho(x,x',t) = \sum_{m,n=1}^{\infty} c_{mn}(t) \psi_n(x) \psi_m(x') $$

If you plug this back in to the von Neumann equation you get that

$$ \frac{\mathrm{d}}{\mathrm{d}t}c_{mn}(t) = \frac{1}{i \hbar} (E_m-E_n) c_{mn}(t) $$

which you can solve:

$$ c_{mn}(t) = c_{mn}(0) e^{-i(E_m-E_n)t/\hbar} $$

where the $c_{mn}(0)$ are set by the initial density matrix.

1) Solution in the energy basis

This is pretty simple actually, just expand the density operator $\hat{\rho}(t)$ as

$$ \hat{\rho}(t) = \sum_{m,n=1}^{\infty} c_{mn}(t) | E_m \rangle\langle E_n | $$

where the eigenvalues and vectors of the Hamiltonian $\hat{H}$ satisfy $\hat{H}| E_n \rangle = E_n | E_n \rangle,~n=1,2,\ldots$ and I assumed that the spectrum of the Hamiltonian is discrete, as is the case for the potential well. Then you substitute this into the von Neumann equation of motion

$$ \frac{\mathrm{d}}{\mathrm{d}t} \hat{\rho}(t) = \frac{1}{i \hbar} [\hat{H},\hat{\rho}]$$

to get

$$ \sum_{m,n=1}^{\infty} \frac{\mathrm{d}c_{mn}(t)}{\mathrm{d}t} | E_m \rangle \langle E_n | = \frac{1}{i \hbar} \sum_{m,n=1}^{\infty} c_{mn}(t) (\hat{H}| E_m \rangle \langle E_n | - | E_m \rangle \langle E_n |\hat{H}).$$

Using $\hat{H}| E_m \rangle = E_m | E_m \rangle$ and $\langle E_n |\hat{H} = \langle E_n | E_n$ you get

$$ \sum_{m,n=1}^{\infty} \frac{\mathrm{d}c_{mn}(t)}{\mathrm{d}t} | E_m \rangle \langle E_n | = \frac{1}{i \hbar} \sum_{m,n=1}^{\infty} c_{mn}(t) (E_m - E_n) | E_m \rangle \langle E_n |.$$

Now you multiply both sides of the equation from the left by $\langle E_{m'} |$ and from the right by $| E_{n'} \rangle$ and supposing the eigenbasis is orthonormal, i.e. $\langle E_n | E_m \rangle = \delta_{nm}$, you end up with

$$\frac{\mathrm{d}c_{m'n'}(t)}{\mathrm{d}t} = \frac{1}{i \hbar} c_{m'n'}(t) (E_{m'} - E_{n'}).$$

From here on the solution is easy, and it is

$$ c_{m'n'}(t) = c_{m'n'}(0) e^{-i(E_{m'}-E_{n'})t/\hbar} $$

where the $c_{m'n'}(0)$ are set by your initial conditions.

2) Reduction to a partial differential equation in the position basis

Now you write down the density operator in position representation as

$$ \hat{\rho}(t) = \int_{-\infty}^{\infty} \mathrm{d}x \int_{-\infty}^{\infty} \mathrm{d} x' \rho(x,x',t) | x \rangle \langle x' | $$ and again substitute it into the von Neumann equation of motion to get

$$ \frac{\mathrm{d}}{\mathrm{d}t}\hat{\rho}(t) = \int_{-\infty}^{\infty} \mathrm{d}x \int_{-\infty}^{\infty} \mathrm{d} x' | x \rangle \langle x' | \frac{\partial}{\partial t}\rho(x,x',t) = \frac{1}{i \hbar} \int_{-\infty}^{\infty} \mathrm{d}x \int_{-\infty}^{\infty} \mathrm{d} x' \rho(x,x',t) (\hat{H}| x \rangle \langle x' | - | x \rangle \langle x' |\hat{H}).$$

Again you use the same trick as before, multiply both sides on the left by $\langle x'' |$ and on the right by $| x''' \rangle$. Hopefully your position basis vectors are normalised as $\langle x | x' \rangle = \delta(x-x')$ and remembering that $\int_{-\infty}^{\infty} f(x) \delta(x-x') \mathrm{d}x = f(x')$, you get

$$ \frac{\partial}{\partial t} \rho(x'',x''',t) = \frac{1}{i \hbar} \int_{-\infty}^{\infty} \mathrm{d}x \rho(x,x''',t) \langle x'' | \hat{H} | x \rangle - \frac{1}{i \hbar} \int_{-\infty}^{\infty} \mathrm{d}x' \rho(x'',x',t) \langle x' | \hat{H} | x''' \rangle.$$

Before proceeding, I relabel variables for slightly easier handling, so we have

$$ \frac{\partial}{\partial t} \rho(x,x',t) = \frac{1}{i \hbar} \int_{-\infty}^{\infty} \mathrm{d}x'' \rho(x'',x',t) \langle x | \hat{H} | x'' \rangle - \frac{1}{i \hbar} \int_{-\infty}^{\infty} \mathrm{d}x'' \rho(x,x'',t) \langle x'' | \hat{H} | x' \rangle.$$

The slightly tricky part now is to figure out what the "matrix elements" $\langle x | \hat{H} | x' \rangle$ are.

To do this, go back to the Schrödinger equation in the form

$$ i \hbar \frac{\mathrm{d}}{\mathrm{d}t} | \Psi(t) \rangle = \hat{H} | \Psi(t) \rangle $$ where the ket $| \Psi(t) \rangle$ is not a wavefunction but a vector in the Hilbert space of the problem. However, it can be expanded in the position basis as $| \Psi(t) \rangle = \int_{-\infty}^{\infty} \mathrm{d}x | x \rangle \langle x | \Psi(t) \rangle$ where the coefficients $\Psi(x,t) = \langle x | \Psi(t) \rangle$ are traditionally known as the wavefunction of the system. Playing the same game as with the von Neumann equation, you can rewrite the Schrödinger equation as

$$ i \hbar \frac{\partial}{\partial t} \Psi(x,t) = \int_{-\infty}^{\infty} \mathrm{d}x' \langle x |\hat{H}| x' \rangle \Psi(x',t).$$ Standard wave mechanics tells you however that that the right-hand side (RHS) must be

$$ \int_{-\infty}^{\infty} \mathrm{d}x' \langle x |\hat{H}| x' \rangle \Psi(x',t) = \left[-\frac{\hbar}{2m} \frac{\partial^2}{\partial x^2} + V(x)\right] \Psi(x,t)$$

which then tells you what happens when you perform integrals of the form $ \int_{-\infty}^{\infty} \mathrm{d}x' \langle x | \hat{H} | x' \rangle f(x') $

Now that we have what we need, go back to the von Neumann equation

$$ i \hbar \frac{\partial}{\partial t} \rho(x,x',t) = \int_{-\infty}^{\infty} \mathrm{d}x'' \rho(x'',x',t) \langle x | \hat{H} | x'' \rangle - \int_{-\infty}^{\infty} \mathrm{d}x'' \rho(x,x'',t) \langle x'' | \hat{H} | x' \rangle $$

where we can now evaluate the first term on the RHS:

$$ i \hbar \frac{\partial}{\partial t} \rho(x,x',t) = \left[ -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right] \rho(x,x',t) - \int_{-\infty}^{\infty} \mathrm{d}x'' \rho(x,x'',t) \langle x'' | \hat{H} | x' \rangle.$$

To evaluate the second term on the RHS, note that $\langle x'' | \hat{H} | x' \rangle = \langle x' | \hat{H}^\dagger | x'' \rangle^*$ where the $*$ denotes complex conjugation and the $\dagger$ symbol means taking the adjoint. The Hamiltonian is self-adjoint (Hermitian), so we have $\langle x'' | \hat{H} | x' \rangle = \langle x' | \hat{H} | x'' \rangle^*$. But the matrix elements are real (a consequence of the time-reversal symmetry of the problem) so nothing really changes, and you can write

$$ i \hbar \frac{\partial}{\partial t} \rho(x,x',t) = \left[ -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right] \rho(x,x',t) - \left[ -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x'^2} + V(x') \right] \rho(x,x',t).$$

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  • $\begingroup$ Thank you so much for the answer. A see that a big problem that I have is that I don't understand how to write the density operator correctly. Could you please explain why you chose that expansion of the density operator in the position basis? $\endgroup$ Nov 29, 2022 at 14:24
  • $\begingroup$ Depends on the level of explanation you need. If you take e.g. real square matrices $\mathbf{A} \in \mathbb{R}^{N \times N}$, they live in an $N^2$ dimensional vector space. A basis $\mathbf{B}_{ij}$ of this space can be formed from any basis $\mathbf{b}_1,\mathbf{b}_2,\ldots,\mathbf{b}_N$ of $\mathbb{R}^N$ by $\mathbf{B}_{ij}=\mathbf{b}_i\mathbf{b}_j^T$ (the outer product of every two basis vectors). So in the end you can write down any matrix as $\mathbf{A}=\sum_{i,j=1}^{N} c_{ij}\mathbf{B}_{ij}$. $\endgroup$ Nov 29, 2022 at 15:41
  • $\begingroup$ If you now consider instead operators $\hat{A}$ acting on an infinite dimensional Hilbert space with basis $|E_k\rangle$, you can write them down as $\hat{A} = \sum_{k,k'=1}^{\infty} a_{kk'} |E_k\rangle \langle E_{k'}|$. If the basis is indexed by a continuous label such as the position basis vectors $| x \rangle$, instead of the sums you write integrals and have $\hat{A} = \int \mathrm{d}x \int \mathrm{d}x' A(x,x') | x \rangle \langle x' |$. This is really heuristic, and to be fair, I never looked up the proper maths behind it, but it does at least work. $\endgroup$ Nov 29, 2022 at 15:46

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