## The variational principle in quantum mechanics, lecture 11

1. Lecture 11: the time-dependent variational principle and mean-field theory

In this and the following lectures we are going to apply the time-dependent variation principle (TDVP) to the variational classes we have met previously, namely, product states for quantum spin systems, mean-field states for bosons and fermions, and matrix product states.

Let’s begin by recalling the TDVP. When we assume that our variational manifold ${\mathcal{M}}$ can be parametrized as

$\displaystyle \mathcal{M}=\{|\psi(\mathbf{z})\rangle,\mathbf{z}\in \mathbb{C}^{n}\}, \ \ \ \ \ (1)$

the TDVP gives the equations of motion

$\displaystyle i\dot{z}^{i}(t) = G^{i\overline{\jmath}}(\overline{\mathbf{z}}(t),\mathbf{z}(t))\langle\partial_{\overline{\jmath}} \psi(\overline{\mathbf{z}}(t))|H|\psi(\mathbf{z}(t))\rangle \ \ \ \ \ (2)$

for ${\mathbf{z}(t)}$ and its complex conjugate.

2. Example 1: a ${1D}$ quantum spin system

In this section we apply the TDVP equations of motion to the description of the dynamics of the ${1D}$ quantum spin systems studied in lecture 4. As before, we assume that these are an array of spin-${\frac12}$ degrees of freedom arranged in a regular one-dimensional lattice. Recall that a convenient basis for a single spin-${\frac12}$ degree of freedom is provided by the eigenstates of the ${s_z}$ spin operator, written ${|0\rangle}$ and ${|1\rangle}$. The hilbert space of a single spin-${\frac12}$ is isomorphic to ${\mathbb{C}^2}$.

The hilbert space ${\mathcal{H}}$ for our collection of ${N}$ such spin-${\frac12}$ degrees of freedom is thus given by

$\displaystyle \mathcal{H} = \underbrace{\mathbb{C}^2\otimes \mathbb{C}^2\otimes \cdots \otimes \mathbb{C}^2}_{N \, {\rm times}}. \ \ \ \ \ (3)$

A general hamiltonian for a quantum spin system has the form

$\displaystyle H = \sum_{j=1}^{N} h_{j,j+1}, \ \ \ \ \ (4)$

where the operator ${h_{j,j+1}}$ acts nontrivially only on spins ${j}$ and ${j+1}$. We’ve identified spins ${N}$ and ${1}$ here, so this systems has periodic boundary conditions.

In this example we apply the TDVP to ${H}$ using as our variational class the set of all states with the form

$\displaystyle \mathcal{V} = \{ |\Psi\rangle \,|\, |\Psi\rangle = |\psi(z)\rangle|\psi(z)\rangle\cdots |\psi(z)\rangle \}, \ \ \ \ \ (5)$

i.e., the set of all symmetric product states. We don’t directly specify the dependence of ${|\psi(z)\rangle}$, except to require that it is analytic. (Here we are using only one complex parameter ${z}$ to specify ${|\psi(z)\rangle}$).

In order to apply the TDVP we’re going to need to understand the tangent space ${T\mathcal{V}}$ to ${\mathcal{V}}$. Thus we differentiate a member ${|\phi(z)\rangle \in \mathcal{V}}$ with respect to ${z}$ to obtain

$\displaystyle |\partial_z \phi(z)\rangle = \sum_{j=1}^N |\psi(z)\rangle_1 \cdots |\partial_z \psi(z)\rangle_j \cdots |\psi(z)\rangle_N. \ \ \ \ \ (6)$

This allows us to calculate the Gram matrix ${G}$ which, for us, is simply a complex number (we only have one variational parameter):

$\displaystyle G(\overline{z}(t), z(t)) = \sum_{j=1}^N \langle \partial_{\overline{z}}\psi(\overline{z})| \partial_z\psi(z)\rangle + \sum_{j\not = k} \langle \partial_{\overline{z}}\psi(\overline{z})|\psi(z)\rangle \langle \psi(z)| \partial_z\psi(z)\rangle, \ \ \ \ \ (7)$

which reduces to (using the symmetry of the product)

$\displaystyle G(\overline{z}(t), z(t)) = N\langle \partial_{\overline{z}}\psi(\overline{z})| \partial_z\psi(z)\rangle + \frac{N(N-1)}{2}\langle \partial_{\overline{z}}\psi(\overline{z})|\psi(z)\rangle \langle \psi(z)| \partial_z\psi(z)\rangle. \ \ \ \ \ (8)$

The other thing we need to calculate is the inner product ${\langle\partial_{\overline{z}} \psi(\overline{z}(t))|H|\psi({z}(t))\rangle}$. This is given, via a similar calculation, by

$\displaystyle N\langle \partial_{\overline{z}} \psi(\overline{z})|\langle \psi(\overline{z})|h|\psi(z)\rangle|\psi(z)\rangle + N\langle \psi(\overline{z})|\langle \partial_{\overline{z}} \psi(\overline{z})|h|\psi(z)\rangle|\psi(z)\rangle +$

$\displaystyle N(N-2) \langle \psi(\overline{z})|\langle \psi(\overline{z})|h|\psi(z)\rangle|\psi(z)\rangle \langle \partial_{\overline{z}}\psi(\overline{z})|\psi(z)\rangle.$

Suppose, for the sake of argument, that ${|\psi(z)\rangle}$ is designed so that it is always normalised (actually this is probably incompatible with the requirement that ${|\psi(z)\rangle}$ is analytic), i.e., ${\langle \psi(\overline{z})|\psi(z)\rangle = 1}$ for all ${z}$, then

$\displaystyle 0 = \langle \partial_{\overline{z}}\psi(\overline{z})|\psi(z)\rangle \ \ \ \ \ (9)$

so that we obtain the simplified equation of motion

$\displaystyle i\dot{z}(t) = \frac{\langle \partial_{\overline{z}} \psi(\overline{z})|\langle \psi(\overline{z})|h|\psi(z)\rangle|\psi(z)\rangle + \langle \psi(\overline{z})|\langle \partial_{\overline{z}} \psi(\overline{z})|h|\psi(z)\rangle|\psi(z)\rangle}{\langle \partial_{\overline{z}}\psi(\overline{z})| \partial_z\psi(z)\rangle}. \ \ \ \ \ (10)$

Otherwise, more generally, we obtain a more complicated expression.

3. Example 2: Bose-Einstein condensates

In this next example it is much more convenient to derive the TDVP directly from a lagrangian representation.

We now apply the TDVP to a hamiltonian ${\widehat{H}}$ modelling a dilute gas of bosons:

$\displaystyle \widehat{H} = \int d^3\mathbf{x}\, \widehat{\psi}^\dag(\mathbf{x}) \left(-\frac{\hbar^2}{2m}\nabla^2+V_{\mbox{ext}}(\mathbf{x})\right) \widehat{\psi}(\mathbf{x}) + \frac{g}2\int d^3\mathbf{x}\, \widehat{\psi}^\dag(\mathbf{x}) \widehat{\psi}^\dag(\mathbf{x}) \widehat{\psi}(\mathbf{x}) \widehat{\psi}(\mathbf{x}), \ \ \ \ \ (11)$

where ${\widehat{\psi}(\mathbf{x})}$ and ${\widehat{\psi}^\dag(\mathbf{x})}$ are the bosonic field annihilation and creation operators.

We use the set of field coherent states

$\displaystyle \{|\phi\rangle\,|\, |\phi\rangle = D(\widehat{\psi}, \phi) |\Omega\rangle, \phi:\mathbb{R}^3\rightarrow \mathbb{C}\}, \ \ \ \ \ (12)$

where

$\displaystyle D(\widehat{\psi}, \phi) \equiv e^{\int d^3\mathbf{x} \, \phi(\mathbf{x})\widehat{\psi}^\dag(\mathbf{x}) - \overline{\phi}(\mathbf{x})\widehat{\psi}(\mathbf{x})}. \ \ \ \ \ (13)$

Recall that a field coherent state obeys

$\displaystyle \widehat{\psi}(\mathbf{x})|\phi\rangle = \phi(\mathbf{x})|\phi\rangle \ \ \ \ \ (14)$

and

$\displaystyle \frac{\partial\widehat{\psi}(\mathbf{x})}{\partial x}|\phi\rangle = \frac{\partial\phi(\mathbf{x})}{\partial x}|\phi\rangle. \ \ \ \ \ (15)$

We now set up the lagrangian for the Schrödinger equation:

$\displaystyle L(\overline{\phi}(x,t),\phi(x,t))=\frac{i}{2}\langle\phi(t)|\partial_t{\phi}(t)\rangle-\frac{i}{2}\langle\partial_t{\phi}(t)|\phi(t)\rangle-\langle\phi(t)|H(t)|\phi(t)\rangle. \ \ \ \ \ (16)$

Using

$\displaystyle \langle\phi(t)|\partial_t{\phi}(t)\rangle = \int d^3\mathbf{x}\, \overline{\phi}(\mathbf{x}) \partial_t\phi(\mathbf{x}), \ \ \ \ \ (17)$

$\displaystyle \langle\partial_t\phi(t)|{\phi}(t)\rangle = \int d^3\mathbf{x}\, (\partial_t\overline{\phi}(\mathbf{x},t)) \phi(\mathbf{x},t), \ \ \ \ \ (18)$

and

$\displaystyle \langle \phi|\widehat{H}|\phi\rangle = \int d^3\mathbf{x}\, \overline{\phi}(\mathbf{x}) \left(-\frac{\hbar^2}{2m}\nabla^2+V_{\mbox{ext}}(\mathbf{x})\right) \phi(\mathbf{x}) + \frac{g}2\int d^3\mathbf{x}\, |\phi(\mathbf{x})|^4, \ \ \ \ \ (19)$

we can write out the lagrangian:

$\displaystyle L= \int \frac{i}{2}\overline{\phi}(\mathbf{x}) \partial_t\phi(\mathbf{x}) - \frac{i}{2}(\partial_t\overline{\phi}(\mathbf{x},t)) \phi(\mathbf{x},t) + \overline{\phi}(\mathbf{x})\left(-\frac{\hbar^2}{2m}\nabla^2+V_{\mbox{ext}}(\mathbf{x})\right) \phi(\mathbf{x}) + \frac{g}2 |\phi(\mathbf{x})|^4 \, d^3\mathbf{x}. \ \ \ \ \ (20)$

To apply the TDVP we need to extremise ${L}$, which we do using variational calculus: the Euler-Lagrange equations give

$\displaystyle 0 = \frac{\delta}{\delta \overline{\phi}(\mathbf{x})}\langle \phi|\widehat{H}|\phi\rangle, \ \ \ \ \ (21)$

which reduce to

$\displaystyle -i\partial_t\phi(\mathbf{x},t) = \left(-\frac{\hbar^2}{2m}\nabla^2+V_{\mbox{ext}}(\mathbf{x})\right) \phi(\mathbf{x}, t) + g |\phi(\mathbf{x},t )|^2 \phi(\mathbf{x}, t). \ \ \ \ \ (22)$

This is the (time-dependent) Gross-Pitaevskii equation (GPE) which, when ${V=0}$, is also known as the nonlinear Schrödinger equation.