You can look up specific dates on Twitter by using the "Advanced Twitter Search" form on the Twitter website. You do not need a Twitter account to search tweets. When you perform a Twitter search for the specific time frame you have in mind, all tweets in that range will display in your browser. Searching Twitter by date is especially helpful when looking for a specific tweet without being able to recall the exact wording or phrasing of the tweet.
Go to the advanced search page at search.twitter.com/advanced.
Scroll down the page until you see the "Dates" section of the form. Two boxes appear in that section.
Type a date in the form of YEAR-MO-DA (year, month, date) in the box labelled "Since this date" to specify the earliest specific date you want to search on Twitter.
Type a date in the form of YEAR-MO-DA (year, month, date) in the box labelled "Until this date" to specify the latest specific date you want to search on Twitter.
Scroll down to the bottom of the form and click "Search" to display all tweets between the dates you specified in the previous steps.
You can also fill in any other boxes -- e.g., keywords, keyword exclusions, originating Twitter users, location and so on -- on the form to narrow or broaden your search.