Learn Russian: Чем больше мы изучаем атом, тем больше сомневаемся. - The more we study the atom, the more we doubt.

The phrase чем больше ..., тем больше ... is the Russian equivalent of 'the more..., the more...'. Grammatically, чем (the instrumental case of что) introduces the first action, and тем (the instrumental of то) introduces the proportional result. You can use this pattern with any comparative adverbs or adjectives.

Why is the subject мы dropped in the second half of the sentence before сомневаемся?

Russian frequently drops the subject in the second clause of a compound sentence if it remains identical to the first. The first-person plural ending of сомневаемся makes it completely unambiguous that мы is still the subject.

Why doesn't the direct object атом change its ending here after the verb изучаем?

It actually is in the accusative case here! Because атом is a masculine inanimate noun, its accusative form is identical to its nominative form. If the object were an animate masculine noun, like a student (студент), it would take the animate accusative ending: изучаем студента.

The verb сомневаемся is reflexive. If we wanted to expand the sentence to say 'we doubt it', how would we connect the object?