I am truly sorry about your mother's Alzheimers. I know from personal experience how devastating a disease it is. Soothing and calming effects have been reported from using essential oils but the inevitable changes to the brain won't be affected in the long run. Having a few plants in the house won't make any difference. However, your mother might enjoy the scent or helping to look after the plants.

Start with keeping in mind that Rosemary comes from the rocky hillsides of the Mediterranean, so it needs as much light as possible and extremely fast draining gritty soil. It is also a substantial shrub so it needs a fair sized container. Indoor lighting is insufficient unless you get grow lights. I can't advise on those since I have never used them. While it is in the house it needs to be right up against the biggest windows you have.

Get some mix suitable for cacti, plant them up in larger pots, put them in the brightest possible position in your home and put them outside as soon an as you can. Rosemary is not tropical and in a pot can take temperatures to 0c once hardened off to outside conditions. In the ground it can survive down to around -5 ish. BTW putting stones in pots for drainage is a long believed myth. Don't bother. Just make sure you use a free draining mix and that the pot has drainage holes.

Regarding the scent. Unless they are in baking sun they release no scent at all. You need to rub the foliage to smell it. Try it with the plants you have.

I suggest you do a lot more searching about Rosemary in cooking. I grow it and use it by the branch solely because it is delicious. I don't expect it to stop me getting Alzheimers if that is my fate. You can also search this forum using the search box at the top. Rosemary is asked about frequently.