Pandas dataframe apply function to entire column

I have some functions that act on a list and return a list. I would like to create a column on a Pandas Dataframe such that the new column is the list returned by one of the functions acting on some other column of the dataframe.

In python-like pseudocode:

def function(parameter, list):
    ...
    return output_list

df['New Column'] = function(parameter, df['Old Column'])

I have tried different options including something like the code above, using .apply() method and others... with no success.

Is there a way to do this? Thank you!

EDIT: See Brian Pendleton's answer for the solution. Columns in a dataframe are pandas' Series objects. Just have to create a Series out of the desired list.

df['New_Column'] = pd.Series(data=function(parameter,list))