copy the correct file based on date??

$whatdate = (Get-Date).Date

$today = $whatdate.ToShortDateString()

$file = (Get-ChildItem 'C:\pstfiles\*.pst').LastWriteTime.ToShortDateString()

foreach ($pst in $file)  

    {$day = $today -like $file} 

        if ($day -like "TRUE") {Copy-Item c:\pstfiles\$pst "c:\temp\PSTTSP"}

this code executes without error but nothing gets copied.  This is because my $file is a date (7/24/2014).   How do I modify this code so I capture the file name yet still do the date comparison?   

I have 3 files in my folder.  File1 modified today.  File2 and File3 modified yesterday.  When when I run my code I want to to see that File1 was modified today and then copy File1.pst to my target location.   Sounds easy but proving to be hard.

mqh7