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.

The pdf version of these notes can be found here.

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.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: