If you are looking to use AJAX in WP. Please follow the below example.

var stateName = $('#state').val();
$.ajax({
      type: "POST",
      url: "<!--?php echo admin_url( 'admin-ajax.php' ); ?-->",
      data: { 'action':'custom_dropdown_city','stateName' : stateName}, 
  }).done(function(results) {
  $("#city_id").html(results);
  });

OPEN function.php under your theme folder

function custom_dropdown_city_callback(){    
    global $wpdb;  
    $userid = get_current_user_id();
    $state = $_REQUEST['stateName'];
    $sql = "SELECT `wa`.`city` from wp_connections wc join wp_connections_address wa on `wc`.`id` = `wa`.`entry_id` where `wc`.`owner` ='".$userid."' and `wc`.`deleted` = '0' and `wa`.`state` like'%".$state."%'"; 
    $results = $wpdb->get_results($sql);
    $num = count($results);
    if($num > 0){
      $dd = "<select name="cityName" id="cityName" class="form-control">";
        for($i=0; $i&lt;$num; $i++){
            $dd.="<option value=".$results[$i]->city.">".$results[$i]-&gt;city."</option>";
        }
    }else{
        $dd.="<option></option>";
    }
    $dd.="</select>";
    echo $dd; 
    exit;
}

add_action('wp_ajax_custom_dropdown_city','custom_dropdown_city_callback');
add_action('wp_ajax_nopriv_custom_dropdown_city','custom_dropdown_city_callback');