How can hybrid orbitals be equivalent? How can hybrid orbitals be equivalent? Take, for example, $\mathrm{sp^3}$ hybridization. The wave functions of each hybrid orbitals are given by:

\begin{align}
\psi_1 &amp;= \psi_s + \psi_{p_x} + \psi_{p_y} + \psi_{p_z}\\
\psi_2 &amp;= \psi_s - \psi_{p_x} + \psi_{p_y} - \psi_{p_z}\\
\psi_3 &amp;= \psi_s - \psi_{p_x} - \psi_{p_y} + \psi_{p_z}\\
\psi_4 &amp;= \psi_s + \psi_{p_x} - \psi_{p_y} - \psi_{p_z}\\
\end{align}
No one of them has same linear combination. So, how can they be equivalent? I am not getting the sense.
The standard set of orbitals (s, p, d, f, ...) form an orthogonal basis set, such that you can describe any continuous function as a linear combination of them.  Now, this also means that you can take some subset of them, and make a new set of orthogonal functions, by appropriate normalized linear combinations, that will contain the full symmetry of the subset chosen. The $sp^{3}$ hybridization is just such a set of combinations of $s$ and $p$ orbitals, chosen to make a set of equivalent orbitals emphasizing a mixed symmetry. 

Note that the standard orbitals are not the only orthogonal basis set that exists - it just happens to be the one that nicely gives solutions to the Schrodinger equation separating out $n$ and $l$. One could use spherical Bessel functions, or any number of other orthogonal basis sets, but it would not be as clean of a solution just because of the form of the differential equation under consideration.