Should I publish this?
As it is mentioned in the comments, we have no access to your paper to determine whether it is a perfect paper to be published or the content is of value to be published. Also, reviewing papers are off-topic to this website.

But, if you are looking for answers regarding [you are] a Junior year student and I did this work without any adviser; my answer will be YES.

There is no obligation for anybody to send his paper for a journal to be peer-reviewed or even to publish his research. The content of the paper is more important than the education level of their writers.

However, I would recommend you to show your research paper to a faculty member. He will review your paper scientifically besides to the review he will do on the academic writing of your paper.

Also, if you want to apply for a masters or PhD program or even if you are seeking for a job; you will need recommendation letters from the faculty members (and showing your paper to a professor is one way to show that you are eligible to get one). The professor whom you will show your paper to may be so glad to write one for his smart student who has worked upon Schrödinger's equation.