"wonder if" or "don't know if "

[YOSHITAKEHORI]

Could anyone answer the following question?
What is the difference between:
(a) I wonder if he is at home.
and
(b) I don't know if he is at home.
Thank you!

[Mister Micawber]

.
The first is actually a sort of question-- a solicitation for someone else to respond with the required information.

The second is a simple statement of the extent of the speaker's knowledge.
.

[Casiopea]

Additionally,

[a] I wonder if the following is true or false: he is home.
[b] I don't know the following information: whether he is home or not.

[YOSHITAKEHORI]

Mr Micawder, Casiopea,

Thank you for your replies. Your replies are quite helpful for me.

[Casiopea]

You're most welcome, Yoshitakehori.