There are psychological effects of breaking up with someone...whether you are the "dumper" or the "dumpee."
When a serious relationship ends and it was either unwanted or not expected, the initial response is most often one of panic and confusion, because a sudden breakup is similar to other traumatic events a person can experience (where something familiar and trusted is taken from them, often without warning.)
If the Break Up Was Unwanted (The Dumpee Experience):
The most common psychological effects experienced by the person being dumped are:
Pain
Obsession/Rumination
Stress Response
Identity Shift
Pain: The psychological pain experienced when we are rejected, betrayed, or abandoned is very real. The same part of the brain that processes physical pain is activated when the emotional pain of a breakup is felt, and the person feels, behaves, and reacts in a similar way as someone who is in a great deal of physical pain. It may be that people who say it feels like their heart has been broken are describing a real physically painful sensation. For adolescents, in particular, breakups can precede the development of major depression, partly because they may not yet have the life skills and experience to manage the psychological pain associated with the end of a significant relationship.
Obsession/Rumination: Because romantic love actually activates the part of the brain which is associated with cravings for food and drugs, a similar experience of craving and withdrawal is to be expected following a breakup. The person experiencing a breakup can't stop thinking about their ex and their past, the "good times," going over and over what went wrong and what they might have been able to do to prevent the breakup. Seeking to reconnect with the ex-partner or continue to want to be some part of their lives is normal, and can include everything from the classic "drunk texting the ex" to the more dangerous forms of obsession and criminal stalking behavior.
Stress Response: The person is attempting to come to terms with a traumatic event, the end of an important relationship. Like other traumatic events, a person can respond with feelings of shock, being in denial, attempts to bargain with the ex to reconsider, feeling angry and sad, grieving, and eventually coming to accept that it's over.
The brain is wired to perceive disconnection from sources of love and attention as highly distressing. To be dumped on the savannah by your tribe, by your mother upon whom you depend completely, or by your partner whom you trust is psychologically scary, painful, and stressful.
As the person is attempting to cope, all of the normal responses to being in a high-stress state can occur. The person's appetite and sleep, ability to pay attention and concentrate, energy and motivation, and immune system can all be affected.
Identity Shift: The person experiences an immediate shift in identity from being a part of a couple to being single again. Rapid shifts in identity cause disorientation for most people, requiring time and emotional and cognitive processing to reorient to their new identity. There can be a loss of status, home, friends, time with children, extended family, places of worship, financial resources, and other changes and losses that must be dealt with, but are not often anticipated. Above all, the loss of "what might have been" must be grieved to acceptance.
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Many people think there are only psychological effects for the person being dumped when a relationship ends. Although the pain is different for the person who makes the decision to call it quits, it does exist and is often no less painful.
If the Break Up Was Wanted (The Dumper Experience):
The most common psychological effects experienced by the dumper are:
Dread/Anticipating Loss
Guilt
Isolation
Grief Postponement
Dread/Anticipating Loss: The person who makes the decision to end a relationship often (and hopefully) does not do so impulsively, but rather after long and careful thought. During this time, where the partner may or may not be aware of the contemplation to end the relationship, the person often experiences a great deal of anxiety and dread as they consider ending the relationship and all of the potential consequences involved. Because the dumper knows the end may be coming, there is less shock and trauma involved in the ending, but the stress of contemplating and disconnecting from a loved one is often no less stressful.
This process of considering whether to end a relationship can take weeks to decades, during which time this stress and dread anticipating the end can be occurring. Many times once the dumper announces that it's over, it may appear they are able to move on very quickly, when in reality they have processed through many of the same kinds of emotions and grieved the end of the relationship for a long time already. The dumpee is just beginning that process when the relationship ends.
Guilt: Many times the person who ends a relationship feels intense guilt over causing harm to someone they care about. They do not want to cause harm, but it is unavoidable if they have determined for whatever reason that the relationship is no longer tenable or healthy for them. If there were a way to end the relationship without anyone being hurt, they would do it. Often it is not possible, so the dumper has to be willing to accept that they have caused another person they care about to suffer. A sensitive, caring person is going to feel bad about this and have to live with this.
Isolation: No one is going to be crying tears for the dumper, however, no matter how bad they feel about ending the relationship. The dumper is often perceived as the "bad guy" for leaving the dumpee feeling hurt and abandoned. Friends and family members often take sides, and the dumper is often seen as obviously strong enough to not need support if they were strong enough to leave, so it is usually given instead to the person who has been left behind, who is in shock.
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A similar shift in identity occurs for the dumper as for the dumpee, who must also navigate their surroundings when what is private becomes public, deciding whether and what to share about the reasons for leaving their partner. Although there are always two sides to every breakup, and partners share responsibility for their relationship problems, the "spin" about a breakup will likely become an oversimplified cliche that never adequately sums up the reality. The dumper will often be in the position of having to decide whether and how to defend their decision to end their relationship to outsiders. Refusing not to comment rarely garners support and often increases feelings of isolation.
Grief Postponement: All of the eventual losses and changes in lifestyle that can result from a breakup (home, kids, family, friends, financial resources, etc) are experienced by the dumper as well as the dumpee, but cannot be openly complained about or openly grieved. The dumper often does not feel they are entitled to grieve those losses, because they were the one who wanted it to end. Grieving the end of the relationship can become complicated, because some losses cannot be grieved ahead of time, but by the time they are being experienced, that grief is not supported or allowed.
Although the initial psychological effects of significant relationship breakups are almost always bad and painful, regardless of who initiates the end of a relationship, the long-term outcome for most people is positive. Most people not only survive breakups, but learn things about themselves, others, and relationships that are helpful for the future. If psychological pain does not appear to be lifting after a few weeks, or a serious decline in the ability to function occurs after a breakup, a person should seek professional counseling.
To read more about Breaking Up:
Breakups aren't all bad: Coping strategies to promote positive outcomes
The Neuroscience of Relationship Breakups
Brain treats rejection like physical pain say scientists
The Science Of The Broken Heart
This Is What Happens To Your Body After A Breakup [VIDEO]
My answer to the question on Quora: What are the psychological effects of breaking up with someone?