Please take a look at this article which explains the head covering from the Quran. Its a short read and to the point.
The following is a summary from one of the Chapters from the book 'Woman in the mirror of Majesty and Beauty' by Ayatullah Jawadi Amuli (Page 350-352).
Hijab in the eyes of the Qur'an is that a woman must completely comprehend that her hijab is not related to her herself for her to say 'I've waived my own right'. Similarly the woman's hijab is not related to her husband so that he may say 'I accept' and neither is the hijab in regards go her family so that they can give their acceptance. The hijab of a woman is 'Haqq Ilahi' (God's right). The sanctity of a woman is not limited to herself nor to her husband or brother or children. Even if they all give their acceptance (in the woman waiving the right of her sanctity) the Qur'an will not accept because the sanctity and respectability of a woman is regarded as Haqq Ilahi and God has created a woman as a being possessing affection so she becomes a teacher of tendresse amd brings a message of affection. If a society leaves this teaching of affection and tendresse and runs behind the base animal desires and lust then they will fall into the same trap that the west fell into.
It becomes apparent that the Ismat of a woman is Haqq Ilahi and has no relation to anyone else. Everyone within a family structure but most importantly the woman is the caretaker of Gods trust. The woman is discussed as the caretaker of Haqq Ilahi in the Qur'an meaning this status, sanctity and respect which is alone God's right, He has given it to the woman and said 'take this right of mine and safeguard it as an Amanat (trust).' A society in which the Qur'an is the Hakim is an affectionate society and the secret to this is that half the society are teachers of affection who are known as mothers. Whether we like it or not, understand it or not, the principles of a family provide the lessons in tendresse and this plays a role in every issue (aspect of life).
Hence there is no difference between men and women in every dimension (of life) for reaching perfection as long as it is based on the Qur'anic teachings meaning the same way the Qur'an has placed together perfection, Hijab, thought and chastity, we must aggregate Hijab and books (studies).
The grandeur of a women is in this
«أن لا یرین الرّجال ولا یراهنَّ الرّجال»
'She does not see a Non Mahram and he should not see her.'
When the Qur'an speaks about Hijab it introduces it asrespect and reverence for the woman such that no one should see her in the light of their animalistic tendencies. The philosophy regarding the Hijab in the Qur'an is
(ذلك أدنی أن یعرفن فلا یؤذین)
Surah Ahzab: verse 59
'that they may be known, and thus they will not be given trouble;'
Because they are the embodiment of sanctity and chastity in a society and are deemed honorable.