Let us say there is an adult male, adult female, a boy, and a girl. They experience time-travel.
As a psychologist, what would you examine before and after the travel to ensure that psychologically, each individual is "fine"?
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I'd like a little more information, such as what you mean by "fine."
Do you mean if they're affected psychologically by seeing something traumatic? If I saw the Black Plague, I would not be fine. In that case, I think standard post-traumatic stress examination would help.
If you mean if time travel is affecting them (depending on your story, since we have no real-world basis), then I think asking about nightmares, cognitive disorders (does the person still know what time they're in, can they tell fantasy from reality, etc), and other psychological tests would suffice.
If they're traveling to rather traumatic time periods, they would need long-term watch. It's sort of like Navy SEALs: they have examinations often to make sure they're still in shape.
In addition to what Krakoom said, I might also want to examine their memories in depth before and after the travel, just to be sure that these are the same individuals who I sent out in the first place (as opposed to some alternate universe people). Depending on the mechanics of your time travel, it might be guaranteed that they are from an alternate universe, in which case I would want to make sure they aren't evil or something; but if time travel is a relatively new technology in your world then the psychologist might not know what to expect.
Modern psychologists are finding that culture plays a significant role in what constitutes healthy behavior (for example, arranged marriage is normal in some societies, and not psychologically damaging because they grew up expecting it, but it would be harmful to forcefully subject an American to that if they expected something else). If they went to the past, then I would definitely want to find out about their ideology at large, in hopes of finding out just how much they may have changed the future (they might have political ideas significantly different from the ones that I would expect them to have, if they accidentally changed the government, for example). I would want to find out how much different their original timeline was from my "altered" future, because there's no way for me to know anything about the "original" timeline they came from without having experienced it myself.
If they went to the future and came back, I would want to ensure that they hadn't been captured and subjected to any kind of brainwashing, so I would monitor them in a comfortable but controlled environment for a few days, and have regular meetings with them to conduct psychological examinations for PTSD, consistency in their stories about the future, and consistency in their statements about their plans for their own immediate future, and consistency in their feelings about past events.
If I wasn't a part of the group which sent them to time travel in the first place, I would have a lot of other questions to ask them, but would specifically want to know about their mental health in the ways Krakoom mentioned above.
what would you examine before and after the travel to ensure that psychologically, each individual is "fine"?
You get a blank piece of paper, look at it, and write NOT FINE on it.
You do not need to examine the time travellers to know that they are not fine. The cultural differences are just too big for them to function normally.
To put a real life example, North Korean defectors to the West need a lot of help to get integrated in South Korean society, and are prone to have mental health issues. Some of them even return North. And, for all the shortcomings of the NK regime, its citizens seem to have familiarity with modern technology, science, etc.
Now, imagine the shock for someone coming from the Middle Ages and who did not ever dreamt of seeing someone else talking from "a box" of each house, or a car...