Is it possible to get from Paris to Lucerne in one day using the train? Any responses would be greatly appreciated.

Sure you can do it easily.
The TGV to Zürich from Gare de Lyon; change for Luzern.
That trip takes 7 hour, 5 minutes with one change.
If you want to make two changes, one at Neuchatel and one at Olten, you can do it in 6 hours and 21 minutes. The changes are at small stations, and minimal delay.
The TGV is a little more expensive, but it is quickler even though it not really a TGV after Frasne near the border.
There are numerous connections from Paris Est as well via Basel. All told there must be 7 or 8 or more connections during the day.

You can look these up easily at the SBB web site.

Be aware that the TGV service Paris-Basel does not always run on Saturdays in Summer.

Also the direct connection, Paris to Bern, that leave Gare de Lyon at 7:44 does not run on Saturday. The route is first to Lausanne and then
Bern. If you want to travel on Saturday, there is a need to check the connections.
Your best be on Saturdays is to take the connection through Lausanne. The train leave Gare de Lyon at 7:44 and arrives in Lausanne less than 4 hours later.
The train to Luzern goes through Bern, but you don't need to change.

I ran into that Saturday schedule last year. I think I was forgetting on purpose. It made me change my plans.

And I should proof read my postings.

Best bet and a couple of others need to be interpolated.

Just in case you'd like to consider a night train, there's one that leaves Paris at 22:40 and arrives in Zurich at 6:20 a.m. From there it's less than an hour to Lucerne. J.

Thank you so much for the replies. Just one question, though. By taking the night train to Switzerland from Paris, would I be missing out on any spectacular or beautiful scenery?

There's beautiful scenery along every railroute, but I think it only gets better when you head south from Zurich. You have not said anything about your plans once you get there. Will you have a few days to really explore the beauty of Switzerland? J. and p.s. you can still look out the windows till dark and wake up with the daylight. Of course, you'll be pretty tired your first day in Switzerland.

I'll be landing in Paris and staying there for five days then on to Lucerne for half a day then on to Munich and Budapest.

Well at least the nighttrain buys you an extra half-day somewhere, and I suppose if you manage a first-class sleeper you'll be well rested. J.

If you take the train from Gare de Lyon in Paris to Berne, the scenery is rather bland because most of it is flat. The TGV route as far as Frasne, which is almost to the border, is over flat country. I am not sure how fast the train goes, but I believe 300 KPH.

I am not one to think that a lot of beautiful scenery can be seen from the train. The tracks are generally on the lowest ground. Even the train ride from Luzern to Meiringen, where the route uses the Brünig Pass is not overwhelming. There is one place immediately before the descent of the pass that a train passenger can catch a glimpse of the valley. I have driven over the pass several times, too. There is a pretty view of the lake at Sarnen.

If you want awesome scenery, ride the gondola from Grindelwald to First.
It is something else. Also, the crest of the Männlichen, that towering ridge between Lauterbrunnen and Grindelwald, had some viewpoints that are incredible.

If you go to via Bern, the land around Neuchatel is flat, but pretty.
Bern to Zürich is a relatively calm route.
The tracks are in deep cuts and buildings block the views in most of the towns.

I think a trip up to the top of Pilatus would be good. You can see out quite well. In Luzern, the transportation museum is a class act. The railroad section is interesting, and the displays on mountain lifts is fascinating. The Swiss have all kinds of names for the different contraptions:
Luftseilbahn, which is large car suspended from a cable.
Gondolbahn, which is features a line of cars with 6 passenger cabins.
Drehtseilbahn - a car pulled up and down a steep slope by a cable
Sesselbahn - an oper air bench hung from a cable. Much like a skilift.
Zahnradbahn = the mountain lines with a middle rail for the wheel of the teeth to grip as the train goes up and down hill.

Thank you once more for all your great help, especially since this will be my first trip to Europe.