How can I make a data binding update as soon as a new character is typed in a TextBox?

I’m learning about bindings in WPF and now I’ve become stuck on a (hopefully) simple matter.

I have a simple FileLister class where you can set a Path property, and then it will give you a listing of files when you access the FileNames property.
Here is that class:

class FileLister:INotifyPropertyChanged {
    private string _path = "";

    public string Path {
        get {
            return _path;
        }
        set {
            if (_path.Equals(value)) return;
            _path = value;
            OnPropertyChanged("Path");
            OnPropertyChanged("FileNames");
        }
    }

    public List<String> FileNames {
        get {
            return getListing(Path);
        }
    }

    private List<string> getListing(string path) {
        DirectoryInfo dir = new DirectoryInfo(path);
        List<string> result = new List<string>();
        if (!dir.Exists) return result;
        foreach (FileInfo fi in dir.GetFiles()) {
            result.Add(fi.Name);
        }
        return result;
    }

    public event PropertyChangedEventHandler PropertyChanged;
    protected void OnPropertyChanged(string property) {
        PropertyChangedEventHandler handler = PropertyChanged;
        if (handler != null) {
            handler(this, new PropertyChangedEventArgs(property));
        }
    }
}

I’m using the the FileLister as a StaticResource in this very simple app:

<Window x:Class="WpfTest4.MainWindow"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    xmlns:local="clr-namespace:WpfTest4"
    Title="MainWindow" Height="350" Width="525">
    <Window.Resources>
        <local:FileLister x:Key="fileLister" Path="d:\temp" />
    </Window.Resources>
    <Grid>
        <TextBox Text="{Binding Source={StaticResource fileLister}, Path=Path, Mode=TwoWay}"
        Height="25" Margin="12,12,12,0" VerticalAlignment="Top" />
        <ListBox Margin="12,43,12,12" Name="listBox1" ItemsSource="{Binding Source={StaticResource ResourceKey=fileLister}, Path=FileNames}"/>
    </Grid>
</Window>

The binding is working. If I change the value in the textbox and then click outside of it, the listbox contents will update (as long as the path exists).

The problem is that I would like to update as soon as a new character is typed, and not wait until the textbox lose focus.

How can I do that?
Is there a way to do this directly in the xaml, or do I have to handle TextChanged or TextInput events on the box?