I'm caught up in this thought. My friend gave me just the melody notes of a song. Here it is,
So, here is the problem: "without having the chords supplied, there is no way of knowing what the chords should be."
For example: The E note that begins the melody might correspond to Em or Cm or A chords.
Is this correct?
You're right that the same melody can be played over a variety of different chord sequences, and that the choice of chords will have a marked effect on how the piece sounds.
One of the modern jazz performers' favourite tricks is to take a well known melody and accompany it with unexpected (yet still musically pleasing) chords.
Note that the key signature for your example is D major (or B minor), having C and F sharpened. So you'd be unlikely to play a Cm chord, since the chord contains a C♮ and the key does not.
The four notes you've provided form too short a melody to make any solid decisions about the right chord. It's very common to use the chord corresponding to the key of the piece, and the snippet ends on a D, so D major will probably work.
Not every note in a melody will correspond to a note that appears in the accompanying chord. For example, if you play a D major chord with the four notes in your example, the F♯ and the D will fit in the chord, and the E and C♯ will not. These can be thought of as "connecting notes".
The answer to your question (as expressed in the title) is "No".
Any melody can be harmonized with a wide variety of chord progressions. You can use this to achieve a variety of sophisticated musical effects. One simple example would be using either a minor chord progression or a major chord progression to harmonize the same melody.
That's what makes music so interesting.
It's a well-established songwriting and compositional technique to take the same melody and harmonize it differently each time the melody is repeated. That is one form of what we refer to in classical music as a "theme and variations".
In jazz, it's call "re-harmonizing" or "using chord substitutions".
Now I don't mean to imply that you can take a given melody and choose chords at random and call it a chord progression that accompanies the melody. There are rules for this, and to learn them, you need to study music theory, or at least to listen to some good music composed by people who know their music theory.
Part Two
Nobody here has yet raised the subjects of harmonic rhythm or counterpoint.
Do you want to compose a different chord for each melody note, so that the chord progression incorporates a distinct bass line? That is called a chorale harmonization in homophonic texture. For example, that is the way that Protestant Christian hymns are often arranged in traditional hymnals.
Or do you want to simply create one chord per measure, or a different chord only on strong beats? In that case certain notes in the melody are treated as notes that are not in the chord, and go by various terms such as non-chord tones or passing tones.
Then there is counterpoint, in which there are multiple melody lines going on at the same time. The song tune is the dominant melody, but there are other lines going on underneath it or around it. In properly composed counterpoint there is still a chord progression that can be perceived and derived by looking at how the notes in the various melody lines line up vertically. Counterpoint is a complex and advanced compositional technique.
In a bar of 4/4 the strongest beats are 1 and 3
If for the bar shown you want one chord for the bar then in the snippet shown you have F# for beat 1 and C# for beat 3. Usually if the tonic note is on a weak beat it is not part of a cadence therefore the tonic chord of D major should not be used.
The best candidate is F# minor (F#, A, C#)
Having said this the example provided is out of context in that it is very short. But be mindful what notes occupy the strongest beats in a bar, and also which bars are strongest in a phrase of music. Usually bars are strong-weak alternatively. This should help with choice of chords.
Taken out of the context of a musical phrase, there is no unambiguously most correct analysis. Everything harmonic is relative to what's being tonicized in a given context, and that can be subjective at times. If you were given both a complete musical phrase to be considered in isolation, and a known harmonic rhythm (which chunks of notes are considered at once in deciding what's a chord), then it's sometimes possible to make a very strong argument for a certain interpretation, and there may even be a "most correct analysis", but tautologies of that kind are pretty uncommon. Usually it's just a collection of either more or less plausible analysis.
Assuming functional harmony to begin with, if harmonic rhythm isn't super explicit - as it would be with very homophonic voicing, e.g. you have only a melody - there's still what's called "implied harmony". One of the absolute most canonical examples of implied harmony recognized by everyone are the Bach two part inventions. Everybody knows and studies these, even non-pianists. Frequently no more than two notes are sounded at once, yet it is still possible to perform harmonic analysis with a high degree of certainty because harmony is implied by what's given very strongly by outlining chords and scales, and through phrasing.
Since music is temporal (your ears gather information over time), it's possible to introduce ambiguity and later resolve that ambiguity by giving your ears more complete information about which functions are currently being expressed. Lots of musical interest comes from your ears trying to solve that puzzle as they gather more information about context. The snippit you've given is too short to draw any conclusions. If we knew something about the key, whether this is at the beginning or end of a phrase, and something about the relationship to other voices, it could be narrowed down substantially, but the possibilities would still be enormous.
Finally, what Wheat Williams said in another answer about there being infinite harmonizations is true in theory, but assumes being able to subdivide indefinitely. He's also spot on in that in practice that's never the case, so there are technically finitely many possible harmonizations, and also in practice, there are further stylistic constraints with regards to structure and voice leading regardless of the genre, because every local small-scale musical descision is contingent upon it's surroundings.
Chord for first note: D-F#-A (Tonic)
Chord for second note: A-C#-E (Dominant)
Chord for third note: A-C#-E-G (Dominant 7th)
Chord for fourth note: D-F#-A-D (Tonic)
