You asked

I feel guilty about having someone else care for my child. Is that wrong?

It does not matter how long you leave your child in someone else's care, be it an hour at time or for a full working day. Do not confuse leaving your child in good care with abandoning your child altogether. If you need to work to provide for your family and child in particular, then you are being a good parent by arranging for the best possible childcare you have access to.
Just because you work, does not make you a bad mother in comparison to stay-at-home mothers. Being a stay-at-home mother does not automatically make you a good mother either. It very much depends on how you spend the time you have with your child. Any guilt you feel should go away over time. As long as you carefully select a day care facility, crèche or nanny, you are doing the best you can as a working mother. If you can be sure that your child is in the care of responsible, loving people, you will feel more secure about your choice and have less guilt and anxiety.

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