> I can't understand why we need to use a ground plane at 1/4 wavelength
> and 1/2 wavelength not needed.

A full dipole has voltage and current waves like this: -

[![enter image description here][1]][1]

Picture from [Wikipedia](https://en.wikipedia.org/wiki/Dipole_antenna).

Now, if you focus your eye at dead centre of the picture you will see that the voltage is always zero volts. This is because a dipole is optimally driven with a balanced voltage source (\$V_O\$). A balanced voltage source is preferred for a dipole antenna. In fact, the voltage and electric field is zero all along the length of the green line below: -

[![enter image description here][2]][2]

This means you can optionally regard that green line as earth (providing the line is driven in a balanced way).

Now if you were to cut the above picture in half you'd have a quarter wave monopole driven with an unbalanced voltage source. An unbalanced voltage source is one that has typically 0 volts on one leg while the other leg does the voltage driving: -

[![enter image description here][3]][3]

And, not surprisingly, it has one half of the impedance presented by the half wave dipole.

  [1]: https://i.sstatic.net/L20T3.gif
  [2]: https://i.sstatic.net/CgmEJ.gif
  [3]: https://i.sstatic.net/1xbyT.jpg