When running a page with HTTPS, you’ll want to also load any external resources such as javascript via HTTPS. A lot of people recommend loading jQuery from the Google CDN via the following javascript script:

<script type="text/javascript">
    var gaJsHost = (("https:" == document.location.protocol) ? "https://" : "http://");
    document.write(unescape("%3Cscript src='" + gaJsHost + "ajax.googleapis.com/ajax/libs/jquery/1.4.2/jquery.min.js' type='text/javascript'%3E%3C/script%3E"));
    document.write(unescape("%3Cscript src='" + gaJsHost + "ajax.googleapis.com/ajax/libs/jqueryui/1.7.2/jquery-ui.min.js' type='text/javascript'%3E%3C/script%3E"));
    document.write(unescape("%3Cscript src='" + gaJsHost + "ajax.googleapis.com/ajax/libs/swfobject/2.1/swfobject.js' type='text/javascript'%3E%3C/script%3E"));       
</script>

This works just fine, however you can let the browser select the protocol depending on the request by the following snippet: