= Solution
{c}
The normalized <Hartree product state> and its equivalent <second quantization> form are
$$
\boxed{\Psi_{\theta,\chi}(\mathbf r_1,\ldots,\mathbf r_N)
=\prod_{j=1}^N\varphi_{(\theta,\chi)}(\mathbf r_j),\qquad
|\theta,\chi;N\rangle=\frac{(\cos\theta\,\hat a_L^\dagger+e^{i\chi}\sin\theta\,\hat a_R^\dagger)^N}{\sqrt{N!}}|0\rangle.}
$$
The product is already symmetric under exchanging <bosons>; no additional $1/\sqrt{N!}$ belongs in the coordinate <wavefunction>. The factor in the <second quantization> expression instead normalizes repeated <creation operators> acting on the <vacuum state>. This is a <fixed-number two-mode condensate>, with every <boson> in the same normalized <orbital>.
Set $c=\cos\theta$ and $s=\sin\theta$. Its <binomial distribution> of occupations gives
$$
\langle\hat n_L\rangle=Nc^2,\quad\langle\hat n_R\rangle=Ns^2,\quad
\langle\hat a_L^\dagger\hat a_R\rangle=Ncs\,e^{i\chi},\qquad
\langle\hat n_L^2+\hat n_R^2\rangle=N+N(N-1)(c^4+s^4).
$$
The <expectation value> of the <Hamiltonian operator> is therefore
$$
\boxed{E(\theta,\chi)=-tN\sin(2\theta)\cos\chi
+w\left[N+N(N-1)\left(1-\frac12\sin^2(2\theta)\right)\right]-\mu N.}
$$
Keeping $N(N-1)$ rather than replacing it by $N^2$ retains the exact pair count of the trial <quantum state>. Equivalently, with $\epsilon=w-\mu$, its interaction contribution is $wN(N-1)(c^4+s^4)$ and its one-particle contribution is $\epsilon N-tN\sin(2\theta)\cos\chi$.
Choose the nonredundant parameter range $0\leq\theta\leq\pi/2$, with $\chi$ modulo $2\pi$. For $t>0$, minimizing over the relative <phase> gives $\chi=0$. Putting $x=\sin(2\theta)\in[0,1]$, the remaining nonconstant <energy> is $-tNx-\tfrac12wN(N-1)x^2$, which decreases with $x$ for the repulsive coupling $w>0$. Hence
$$
\boxed{\theta=\frac\pi4,\qquad\chi=0\pmod{2\pi},\qquad
E_{\min}=-tN+\frac w2N(N+1)-\mu N.}
$$
The optimal <orbital> is the symmetric $\phi_0=(\varphi_L+\varphi_R)/\sqrt2$. Both the <quantum tunnelling> term and the repulsive interaction favor this balanced occupation within the <Hartree product state> family. This result uses a <variational method>: for strong repulsion relative to <quantum tunnelling>, the true many-body <ground state> can have smaller occupation fluctuations than this <fixed-number two-mode condensate>.
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