Query on DATETIME for a date (the whole day)

For a "timestamp with time zone" column, how do I find all rows where the value is within the beginning and ending of a
singledate? 

I've studied:
  http://www.postgresql.org/docs/current/static/datatype-datetime.html
  http://www.postgresql.org/docs/current/static/functions-datetime.html

But I'm overwhelmed. I can think of several complicated ways involving tearing apart dates into string parts to
constructbeginning and ending times of day. Surely there must be a simpler way.  

Something like this imaginary command "DATE_OF" where moment_of_birth_ is a "timestamp with time zone" column to find
NewYear's Day babies: 

  SELECT name_, moment_of_birth_
  FROM baby_
  WHERE DATE_OF(moment_of_birth_) = '2011-01-01';

Perhaps use Postgres' casting ability?

  SELECT name_, moment_of_birth_
  FROM baby_
  WHERE moment_of_birth_::date = '2011-01-01'::date ;

What's the best/simplest/efficient way to do this?

And I'm concerned about local date time. I want to find by the user's local beginning and end of the day, not UTC.

--Basil Bourque