An interesting idea is that the universe could be spontaneously created from nothing , but no rigorous proof has been given .
In this paper , we present such a proof based on the analytic solutions of the Wheeler-DeWitt equation ( WDWE ) .
Explicit solutions of the WDWE for the special operator ordering factor p = -2 ( or 4 ) show that , once a small true vacuum bubble is created by quantum fluctuations of the metastable false vacuum , it can expand exponentially no matter whether the bubble is closed , flat or open .
The exponential expansion will end when the bubble becomes large and thus the early universe appears .
With the de Broglie-Bohm quantum trajectory theory , we show explicitly that it is the quantum potential that plays the role of the cosmological constant and provides the power for the exponential expansion of the true vacuum bubble .
So it is clear that the birth of the early universe completely depends on the quantum nature of the theory .